Marks Standard Handbook for Mechanical Engineers

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Marks Standard Handbook for Mechanical Engineers

Marks' Standard Handbook for Mechanical Engineers Revised by a staff of specialists EUGENE A. AVALLONE Editor Consul

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Marks'

Standard Handbook for Mechanical Engineers Revised by a staff of specialists

EUGENE A. AVALLONE

Editor

Consulting Engineer; Professor of Mechanical Engineering, Emeritus The City College of the City University of New York

THEODORE BAUMEISTER III

Editor

Retired Consultant, Information Systems Department E. I. du Pont de Nemours & Co.

ALI M. SADEGH

Editor

Consulting Engineer; Professor of Mechanical Engineering The City College of the City University of New York

Eleventh Edition

New York Chicago San Francisco Lisbon London Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Madrid

Library of Congress Cataloged The First Issue of this title as follows: Standard handbook for mechanical engineers. 1st-ed.; 1916– New York, McGraw-Hill. v. Illus. 18–24 cm. Title varies: 1916–58; Mechanical engineers’ handbook. Editors: 1916–51, L. S. Marks.—1958– T. Baumeister. Includes bibliographies. 1. Mechanical engineering—Handbooks, manuals, etc. I. Marks, Lionel Simeon, 1871– ed. II. Baumeister, Theodore, 1897– ed. III. Title; Mechanical engineers’ handbook. TJ151.S82 502’.4’621 16–12915 Library of Congress Catalog Card Number: 87-641192

MARKS’ STANDARD HANDBOOK FOR MECHANICAL ENGINEERS Copyright © 2007, 1996, 1987, 1978 by The McGraw-Hill Companies, Inc. Copyright © 1967, renewed 1995, and 1958, renewed 1986, by Theodore Baumeister III. Copyright © 1951, renewed 1979, by Lionel P. Marks and Alison P. Marks. Copyright © 1941, renewed 1969, and 1930, renewed 1958, by Lionel Peabody Marks. Copyright © 1924, renewed 1952 by Lionel S. Marks. Copyright © 1916 by Lionel S. Marks. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 1 0 9 8 ISBN-13: 978-0-07-142867-5 ISBN-10: 0-07-142867-4

The sponsoring editor for this book was Larry S. Hager, the editing supervisor was David E. Fogarty, and the production supervisor was Richard C. Ruzycka. It was set in Times Roman by International Typesetting and Composition. The art director for the cover was Anthony Landi. Printed and bound by RR Donnelley. This book is printed on acid-free paper. The editors and the publisher will be grateful to readers who notify them of any inaccuracy or important omission in this book.

Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGrawHill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

Contributors

Abraham Abramowitz* Consulting Engineer; Professor of Electrical Engineering, Emeritus, The City College of The City University of New York (ILLUMINATION) Vincent M. Altamuro President, VMA Inc., Toms River, NJ (MATERIAL HOLDING, FEEDING, AND METERING. CONVEYOR MOVING AND HANDLING. AUTOMATED GUIDED VEHICLES AND ROBOTS. MATERIAL STORAGE AND WAREHOUSING. METHODS ENGINEERING. AUTOMATIC MANUFACTURING. INDUSTRIAL PLANTS) Charles A. Amann Principal Engineer, KAB Engineering (AUTOMOTIVE ENGINEERING) Farid M. Amirouche Professor of Mechanical and Industrial Engineering, University of Illinois at Chicago (INTRODUCTION TO THE FINITE-ELEMENT METHOD. COMPUTER-AIDED DESIGN, COMPUTER-AIDED ENGINEERING, AND VARIATIONAL DESIGN) Yiannis Andreopoulos Professor of Mechanical Engineering, The City College of the City University of New York (EXPERIMENTAL FLUID MECHANICS) William Antis* Technical Director, Maynard Research Council, Inc., Pittsburgh, PA (METHODS ENGINEERING) Glenn E. Asauskas Lubrication Engineer, Chevron Corp. (LUBRICANTS AND LUBRICATION) Dennis N. Assanis Professor of Mechanical Engineering, University of Michigan (INTERNAL COMBUSTION ENGINES) Eugene A. Avallone Consulting Engineer; Professor of Mechanical Engineering, Emeritus, The City College of The City University of New York (MECHANICAL PROPERTIES OF MATERIALS. GENERAL PROPERTIES OF MATERIALS. PIPE, PIPE FITTINGS, AND VALVES. SOURCES OF ENERGY. STEAM ENGINES. MISCELLANY) Klemens C. Baczewski Consulting Engineer (CARBONIZATION OF COAL AND GAS MAKING) Glenn W. Baggley* Former Manager, Regenerative Systems, Bloom Engineering Co., Inc. (COMBUSTION FURNACES) Frederick G. Baily Consulting Engineer; Steam Turbines, General Electric Co. (STEAM TURBINES) Robert D. Bartholomew Associate, Sheppard T. Powell Associates, LLC (CORROSION) George F. Baumeister President, EMC Process Corp., Newport, DE (MATHEMATICAL TABLES) John T. Baumeister Manager, Product Compliance Test Center, Unisys Corp. (MEASURING UNITS) E. R. Behnke* Product Manager, CM Chain Division, Columbus, McKinnon Corp. (CHAINS) John T. Benedict* Retired Standards Engineer and Consultant, Society of Automotive Engineers (AUTOMOTIVE ENGINEERING) Bernadette M. Bennett, Esq. Associate; Carter, DeLuca, Farrell and Schmidt, LLP Melville, NY (PATENTS, TRADEMARKS, AND COPYRIGHTS) Louis Bialy Director, Codes & Product Safety, Otis Elevator Company (ELEVATORS, DUMBWAITERS, AND ESCALATORS) Malcolm Blair Technical and Research Director, Steel Founders Society of America (IRON AND STEEL CASTINGS) Omer W. Blodgett Senior Design Consultant, Lincoln Electric Co. (WELDING AND CUTTING) B. Douglas Bode Engineering Supervisor, Product Customization and Vehicle Enhancement, Construction and Forestry Div., John Deere (OFF-HIGHWAY VEHICLES AND EARTHMOVING EQUIPMENT) Donald E. Bolt* Engineering Manager, Heat Transfer Products Dept., Foster Wheeler Energy Corp. (POWER PLANT HEAT EXCHANGERS) G. David Bounds Senior Engineer, Duke Energy Corp. (PIPELINE TRANSMISSION) William J. Bow* Director, Retired, Heat Transfer Products Department, Foster Wheeler Energy Corp. (POWER PLANT HEAT EXCHANGERS)

*Contributions by authors whose names are marked with an asterisk were made for the previous edition and have been revised or rewritten by others for this edition. The stated professional position in these cases is that held by the author at the time of his or her contribution.

James L. Bowman* Senior Engineering Consultant, Rotary-Reciprocating Compressor Division, Ingersoll-Rand Co. (COMPRESSORS)

Walter H. Boyes, Jr. Editor-in-Chief/Publisher, Control Magazine (INSTRUMENTS) Richard L. Brazill Technology Specialist, ALCOA Technical Center, ALCOA (ALUMINUM AND ITS ALLOYS)

Frederic W. Buse* Chief Engineer, Standard Pump Division, Ingersoll-Rand Co. (DISPLACEMENT PUMPS)

Charles P. Butterfield Chief Engineer, National Wind Technology Center, National Renewable Energy Laboratory (WIND POWER) Late Fellow Engineer, Research Labs., Westinghouse Electric Corp. (NONFERROUS METALS AND ALLOYS. METALS AND ALLOYS FOR NUCLEAR ENERGY APPLICATIONS) Scott W. Case Professor of Engineering Science & Mechanics, Virginia Polytechnic Institute and State University (MECHANICS OF COMPOSITE MATERIALS) Vittorio (Rino) Castelli Senior Research Fellow, Retired, Xerox Corp.; Engineering Consultant (FRICTION. FLUID FILM BEARINGS) Paul V. Cavallaro Senior Mechanical Research Engineer, Naval Undersea Warfare Center (AIR-INFLATED FABRIC STRUCTURES) Eric L. Christiansen Johnson Space Center, NASA (METEOROID/ORBITAL DEBRIS SHIELDING) Robin O. Cleveland Associate Professor of Aerospace and Mechanical Engineering, Boston University (SOUND, NOISE, AND ULTRASONICS) Gary L. Cloud Professor, Department of Mechanical Engineering, Michigan State University (EXPERIMENTAL STRESS AND STRAIN ANALYSIS) Ashley C. Cockerill Vice President and Event Coordinator, nanoTech Business, Inc. (ENGINEERING STATISTICS AND QUALITY CONTROL) Timothy M. Cockerill Senior Project Manager, University of Illinois (ELECTRONICS) Thomas J. Cockerill Advisory Engineer, International Business Machines Corp. (COMPUTERS) Aaron Cohen Retired Center Director, Lyndon B. Johnson Space Center, NASA; Zachry Professor, Texas A&M University (ASTRONAUTICS) Arthur Cohen Former Manager, Standards and Safety Engineering, Copper Development Assn. (COPPER AND COPPER ALLOYS) D. E. Cole Director, Office for Study of Automotive Transportation, Transportation Research Institute, University of Michigan (INTERNAL COMBUSTION ENGINES) James M. Connolly Section Head, Projects Department, Jacksonville Electric Authority (COST OF ELECTRIC POWER) Alexander Couzis Professor of Chemical Engineering, The City College of the City University of New York (INTRODUCTION TO NANOTECHNOLOGY) Terry L. Creasy Assistant Professor of Mechanical Engineering, Texas A&M University (STRUCTURAL COMPOSITES) M. R. M. Crespo da Silva* University of Cincinnati (ATTITUDE DYNAMICS, STABILIZATION, AND CONTROL OF SPACECRAFT) Richard A. Dahlin Vice President, Engineering, Walker Magnetics (LIFTING MAGNETS) Benjamin C. Davenny Acoustical Consultant, Acentech Inc., Cambridge, MA (SOUND, NOISE, AND ULTRASONICS) William H. Day President, Longview Energy Associates, LLC; formerly Founder and Board Chairman, The Gas Turbine Association (GAS TURBINES) Benjamin B. Dayton Consulting Physicist, East Flat Rock, NC (HIGH-VACUUM PUMPS) Horacio M. de la Fuente Senior Engineer, NASA Johnson Space Center (TRANSHAB) Donald D. Dodge Supervisor, Retired, Product Quality and Inspection Technology, Manufacturing Development, Ford Motor Co. (NONDESTRUCTIVE TESTING) Andrew M. Donaldson Project Director, Parsons E&C, Reading, PA (COST OF ELECTRIC POWER) Joseph S. Dorson Senior Engineer, Columbus McKinnon Corp. (CHAIN) James Drago Manager, Engineering, Garlock Sealing Technologies (PACKING, GASKETS, AND SEALS) Michael B. Duke Chief, Solar Systems Exploration, Johnson Space Center, NASA (DYNAMIC ENVIRONMENTS) F. J. Edeskuty Retired Associate, Las Alamos National Laboratory (CRYOGENICS)

C. L. Carlson*

ix

x

CONTRIBUTORS

O. Elnan* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE. ORBITAL MECHANICS)

Robert E. Eppich Vice President, Technology, American Foundry Society (IRON

AND

STEEL CASTINGS)

C. James Erickson*

Retired Principal Consultant, Engineering Department, E. I. du Pont de Nemours & Co. (ELECTRICAL ENGINEERING) George H. Ewing* Retired President and Chief Executive Officer, Texas Eastern Gas Pipeline Co. and Transwestern Pipeline Co. (PIPELINE TRANSMISSION) Heimir Fanner Chief Design Engineer, Ariel Corp. (COMPRESSORS) Erich A. Farber Distinguished Service Professor Emeritus, Director Emeritus of Solar Energy and Energy Conversion Lab., University of Florida [STIRLING (HOT AIR) ENGINES. SOLAR ENERGY. DIRECT ENERGY CONVERSION] Raymond E. Farrell, Esq. Partner; Carter, DeLuca, Farrell and Schmidt, LLP, Melville, NY (PATENTS, TRADEMARKS, AND COPYRIGHTS) D. W. Fellenz* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE. ATMOSPHERIC ENTRY) Chuck Fennell Program Manager, Dalton Foundries (FOUNDARY PRACTICE AND EQUIPMENT) Arthur J. Fiehn* Late Retired Vice President, Project Operations Division, Burns & Roe, Inc. (COST OF ELECTRIC POWER) Sanford Fleeter McAllister Distinguished Professor, School of Mechanical Engineering, Purdue University (JET PROPULSION AND AIRCRAFT PROPELLERS) Luc G. Fréchette Professor of Mechanical Engineering, Université de Sherbrooke, Canada [AN INTRODUCTION TO MICROELECTROMECHANICAL SYSTEMS (MEMS)] William L. Gamble Professor Emeritus of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign (CEMENT, MORTAR, AND CONCRETE. REINFORCED CONCRETE DESIGN AND CONSTRUCTION) Robert F. Gambon Power Plant Design and Development Consultant (COST OF ELECTRIC POWER) Burt Garofab Senior Engineer, Pittston Corp. (MINES, HOISTS, AND SKIPS. LOCOMOTIVE HAULAGE, COAL MINES) Siamak Ghofranian Senior Engineer, Rockwell Aerospace (DOCKING OF TWO FREEFLYING SPACECRAFT) Samuel V. Glorioso Section Chief, Metallic Materials, Johnson Space Center, NASA (STRESS CORROSION CRACKING) Norman Goldberg Consulting Engineer, Economides and Goldberg (AIR-CONDITIONING, HEATING, AND VENTILATING) Andrew Goldenberg Professor of Mechanical and Industrial Engineering, University of Toronto, Canada; President and CEO of Engineering Service Inc. (ESI) Toronto (ROBOTICS, MECHATRONICS, AND INTELLIGENT AUTOMATION) David T. Goldman* Late Deputy Manager, U.S. Department of Energy, Chicago Operations Office (MEASURING UNITS) Frank E. Goodwin Executive Vice President, ILZRO, Inc. (BEARING METALS. LOWMELTING-POINT METALS AND ALLOYS. ZINC AND ZINC ALLOYS) Don Graham Manager, Turning Products, Carboloy, Inc. (CEMENTED CARBIDES) David W. Green Supervisory Research General Engineer, Forest Products Lab., USDA (WOOD) Leonard M. Grillo Principal, Grillo Engineering Co. (MUNICIPAL WASTE COMBUSTION) Walter W. Guy Chief, Crew and Thermal Systems Division, Johnson Space Center, NASA (SPACECRAFT LIFE SUPPORT AND THERMAL MANAGEMENT) Marsbed Hablanian Retired Manager of Engineering and R&D, Varian Vacuum Technologies (HIGH-VACUUM PUMPS) Christopher P. Hansen Structures and Mechanism Engineer, NASA Johnson Space Center (PORTABLE HYPERBARIC CHAMBER) Harold V. Hawkins* Late Manager, Product Standards and Services, Columbus McKinnon Corp. (DRAGGING, PULLING, AND PUSHING. PIPELINE FLEXURE STRESSES) Keith L. Hawthorne Vice President—Technology, Transportation Technology Center, Inc. (RAILWAY ENGINEERING) V. Terrey Hawthorne Late Senior Engineer, LTK Engineering Services (RAILWAY ENGINEERING) J. Edmund Hay U.S. Department of the Interior (EXPLOSIVES) Terry L. Henshaw Consulting Engineer, Magnolia, TX (DISPLACEMENT PUMPS. CENTRIFUGAL PUMPS) Roland Hernandez Research Engineer, Forest Products Lab., USDA (WOOD) David T. Holmes Manager of Engineering, Lift-Tech International Div. of Columbus McKinnon Corp. (MONORAILS. OVERHEAD TRAVELING CRANES) Hoyt C. Hottel Late Professor Emeritus, Massachusetts Institute of Technology (RADIANT HEAT TRANSFER) Michael W. Hyer Professor of Engineering Science & Mechanics, Virginia Polytechnic Institute and State University (MECHANICS OF COMPOSITE MATERIALS) Timothy J. Jacobs Research Fellow, Department of Mechanical Engineering, University of Michigan (INTERNAL COMBUSTION ENGINES) Michael W. M. Jenkins Professor, Aerospace Design, Georgia Institute of Technology (AERONAUTICS) Peter K. Johnson Consultant (POWDERED METALS) Randolph T. Johnson Naval Surface Warfare Center (ROCKET FUELS) Robert L. Johnston Branch Chief, Materials, Johnson Space Center, NASA (METALLIC MATERIALS FOR AEROSPACE APPLICATIONS. MATERIALS FOR USE IN HIGH-PRESSURE OXYGEN SYSTEMS)

Byron M. Jones*

Retired Assistant Professor of Electrical Engineering, School of Engineering, University of Tennessee at Chattanooga (ELECTRONICS) Scott K. Jones Professor, Department of Accounting & MIS, Alfred Lerner College of Business & Economics, University of Delaware (COST ACCOUNTING) Robert Jorgensen Engineering Consultant (FANS) Serope Kalpakjian Professor Emeritus of Mechanical and Materials Engineering, Illinois Institute of Technology (MACHINING PROCESSES AND MACHINE TOOLS) Igor J. Karassik* Late Senior Consulting Engineer, Ingersoll Dresser Pump Co. (CENTRIFUGAL AND AXIAL FLOW PUMPS) Jonathan D. Kemp Vibration Consultant, Acentech, Inc., Cambridge, MA (SOUND, NOISE, AND ULTRASONICS) J. Randolph Kissell President, The TGB Partnership (ALUMINUM AND ITS ALLOYS) John B. Kitto, Jr. Babcock & Wilcox Co. (STEAM BOILERS) Andrew C. Klein Professor of Nuclear Engineering, Oregon State University; Director of Training, Education and Research Partnership, Idaho National Laboratories (NUCLEAR POWER. ENVIRONMENTAL CONTROL. OCCUPATIONAL SAFETY AND HEALTH. FIRE PROTECTION) Doyle Knight Professor of Mechanical and Aerospace Engineering, Rutgers University (INTRODUCTION TO COMPUTATIONAL FLUID MECHANICS) Ronald M. Kohner President, Landmark Engineering Services, Ltd. (DERRICKS) Ezra S. Krendel Emeritus Professor of Operations Research and Statistics, University of Pennsylvania (HUMAN FACTORS AND ERGONOMICS. MUSCLE GENERATED POWER) A. G. Kromis* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE) Srirangam Kumaresan Biomechanics Institute, Santa Barbara, California (HUMAN INJURY TOLERANCE AND ANTHROPOMETRIC TEST DEVICES) L. D. Kunsman* Late Fellow Engineer, Research Labs, Westinghouse, Electric Corp. (NONFERROUS METALS AND ALLOYS. METALS AND ALLOYS FOR NUCLEAR ENERGY APPLICATIONS) Colin K. Larsen Vice President, Blue Giant U.S.A. Corp. (SURFACE HANDLING) Stan Lebow Research Forest Products Technologist, Forest Products Lab., USDA (WOOD) John H. Lewis Engineering Consultant; Formerly Engineering Staff, Pratt & Whitney Division, United Technologies Corp.; Adjunct Associate Professor, Hartford Graduate Center, Renssealear Polytechnic Institute (GAS TURBINES) Jackie Jie Li Professor of Mechanical Engineering, The City College of the City University of New York (FERROELECTRICS/PIEZOELECTRICS AND SHAPE MEMORY ALLOYS) Peter E. Liley Professor Emeritus of Mechanical Engineering, Purdue University (THERMODYNAMICS, THERMODYNAMIC PROPERTIES OF SUBSTANCES) James P. Locke Flight Surgeon, NASA Johnson Space Center (PORTABLE HYPERBARIC CHAMBER) Ernst K. H. Marburg Manager, Product Standards and Service, Columbus McKinnon Corp. (LIFTING, HOISTING, AND ELEVATING. DRAGGING, PULLING, AND PUSHING. LOADING, CARRYING, AND EXCAVATING) Larry D. Means President, Means Engineering and Consulting (WIRE ROPE) Leonard Meirovitch University Distinguished Professor Emeritus, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University (VIBRATION) George W. Michalec Late Consulting Engineer (GEARING) Duane K. Miller Manager, Engineering Services, Lincoln Electric Co. (WELDING AND CUTTING) Patrick C. Mock Principal Electron Optical Scientist, Science and Engineering Associates, Inc. (OPTICS) Thomas L. Moser Deputy Associate Administrator, Office of Space Flight, NASA Headquarters, NASA (SPACE-VEHICLE STRUCTURES) George J. Moshos* Professor Emeritus of Computer and Information Science, New Jersey Institute of Technology (COMPUTERS) Eduard Muljadi Senior Engineer, National Wind Technology Center, National Renewable Energy Laboratory (WIND POWER) Otto Muller-Girard* Consulting Engineer (INSTRUMENTS) James W. Murdock Late Consulting Engineer (MECHANICS OF FLUIDS) Gregory V. Murphy Head and Professor, Department of Electrical Engineering, College of Engineering, Tuskegee University (AUTOMATIC CONTROLS) Joseph F. Murphy Research General Engineer, Forest Products Lab., USDA (WOOD) John Nagy Retired Supervisory Physical Scientist, U.S. Department of Labor, Mine Safety and Health Administration (DUST EXPLOSIONS) B. W. Niebel* Professor Emeritus of Industrial Engineering, The Pennsylvania State University (INDUSTRIAL ECONOMICS AND MANAGEMENT) James J. Noble Formerly Research Associate Professor of Chemical and Biological Engineering, Tufts University (RADIANT HEAT TRANSFER) Charles Osborn Business Manager, Precision Cleaning Division, PTI Industries, Inc. (PRECISION CLEANING) D. J. Patterson Professor of Mechanical Engineering, Emeritus, University of Michigan (INTERNAL COMBUSTION ENGINES) Harold W. Paxton United States Steel Professor Emeritus, Carnegie Mellon University (IRON AND STEEL) Richard W. Perkins Professor Emeritus of Mechanical, Aerospace, and Manufacturing Engineering, Syracuse University (WOODCUTTING TOOLS AND MACHINES) W. R. Perry* University of Cincinnati (ORBITAL MECHANICS. SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE)

CONTRIBUTORS Kenneth A. Phair Senior Project Engineer, Shaw Stone & Webster (GEOTHERMAL POWER)

Orvis E. Pigg Section Head, Structural Analysis, Johnson Space Center, NASA (SPACE VEHICLE STRUCTURES)

Henry O. Pohl Chief, Propulsion and Power Division, Johnson Space Center, NASA (SPACE PROPULSION)

Nicholas R. Rafferty Retired Technical Associate, E. I. du Pont de Nemours & Co., Inc. (ELECTRICAL ENGINEERING) Rama Ramakumar PSO/Albrecht Naeter Professor and Director, Engineering Energy Laboratory, Oklahoma State University (WIND POWER) Pascal M. Rapier* Scientist III, Retired, Lawrence Berkeley Laboratory (ENVIRONMENTAL CONTROL. OCCUPATIONAL SAFETY AND HEALTH. FIRE PROTECTION) James D. Redmond President, Technical Marketing Services, Inc. (STAINLESS STEELS) Darrold E. Roen Late Manager, Sales & Special Engineering & Government Products, John Deere (OFF-HIGHWAY VEHICLES) Ivan L. Ross* International Manager, Chain Conveyor Division, ACCO (OVERHEAD CONVEYORS) Robert J. Ross Supervisory Research General Engineer, Forest Products Lab., USDA (WOOD) J. W. Russell* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE. LUNAR- AND INTERPLANETARY FLIGHT MECHANICS) A. J. Rydzewski Consultant, DuPont Engineering, E. I. du Pont de Nemours and Company (MECHANICAL REFRIGERATION) Ali M. Sadegh Professor of Mechanical Engineering, The City College of The City University of New York (MECHANICS OF MATERIALS. NONMETALLIC MATERIALS. MECHANISM. MACHINE ELEMENTS. SURFACE TEXTURE DESIGNATION, PRODUCTION, AND QUALITY CONTROL. INTRODUCTION TO BIOMECHANICS. AIR-INFLATED FABRIC STRUCTURES. RAPID PROTOTYPING.) Anthony Sances, Jr. Biomechanics Institute, Santa Barbara, CA (HUMAN INJURY TOLERANCE AND ANTHROPOMETRIC TEST DEVICES) C. Edward Sandifer Professor, Western Connecticut State University, Danbury, CT (MATHEMATICS) Erwin M. Saniga Dana Johnson Professor of Information Technology and Professor of Operations Management, University of Delaware (OPERATIONS MANAGEMENT) Adel F. Sarofim Presidential Professor of Chemical Engineering, University of Utah (RADIANT HEAT TRANSFER) Martin D. Schlesinger Late Consultant, Wallingford Group, Ltd. (FUELS) John R. Schley Manager, Technical Marketing, RMI Titanium Co. (TITANIUM AND ZIRCONIUM) Matthew S. Schmidt Senior Engineer, Rockwell Aerospace (DOCKING OF TWO FREEFLYING SPACECRAFT)

xi

Wiliam C. Schneider Retired Assistant Director Engineering/Senior Engineer, NASA Johnson Space Center; Visiting Professor, Texas A&M University (ASTRONAUTICS)

James D. Shearouse, III Late Senior Development Engineer, The Dow Chemical Co. (MAGNESIUM AND MAGNESIUM ALLOYS)

David A. Shifler Metallurgist, MERA Metallurgical Services (CORROSION) Rajiv Shivpuri Professor of Industrial, Welding, and Systems Engineering, Ohio State University (PLASTIC WORKING OF METALS)

James C. Simmons Senior Vice President, Business Development, Core Furnace Systems Corp. (ELECTRIC FURNACES AND OVENS)

William T. Simpson Research Forest Products Technologist, Forest Products Lab., USDA (WOOD)

Kenneth A. Smith Edward R. Gilliland Professor of Chemical Engineering, Massachusetts Institute of Technology (TRANSMISSION OF HEAT BY CONDUCTION AND CONVECTION)

Lawrence H. Sobel* University of Cincinnati (VIBRATION OF STRUCTURES) James G. Speight Western Research Institute (FUELS) Robert D. Steele Project Manager, Voith Siemens Hydro Power Generation, Inc. (HYDRAULIC TURBINES)

Stephen R. Swanson Professor of Mechanical Engineering, University of Utah (FIBER COMPOSITE MATERIALS)

John Symonds* Fellow Engineer, Retired, Oceanic Division, Westinghouse Electric Corp. (MECHANICAL PROPERTIES OF MATERIALS)

Peter L. Tea, Jr. Professor of Physics Emeritus, The City College of the City University of New York (MECHANICS OF SOLIDS)

Anton TenWolde Supervisory Research Physicist, Forest Products Lab., USDA (WOOD) W. David Teter Retired Professor of Civil Engineering, University of Delaware (SURVEYING) Michael C. Tracy Rear Admiral, U.S. Navy (MARINE ENGINEERING) John H. Tundermann Former Vice President, Research and Technology, INCO Intl., Inc. (METALS AND ALLOYS FOR USE AT ELEVATED TEMPERATURES. NICKEL AND NICKEL ALLOYS)

Charles O. Velzy Consultant (MUNICIPAL WASTE COMBUSTION) Harry C. Verakis Supervisory Physical Scientist, U.S. Department of Labor, Mine Safety and Health Administration (DUST EXPLOSIONS)

Arnold S. Vernick Former Associate, Geraghty & Miller, Inc. (WATER) Robert J. Vondrasek* Assistant Vice President of Engineering, National Fire Protection Assoc. (COST OF ELECTRIC POWER)

Michael W. Washo Senior Engineer, Retired, Eastman Kodak, Co. (BEARINGS

WITH

ROLLING CONTACT)

Larry F. Wieserman Senior Technical Supervisor, ALCOA (ALUMINUM AND ITS ALLOYS) Robert H. White Supervisory Wood Scientist, Forest Products Lab., USDA (WOOD) John W. Wood, Jr. Manager, Technical Services, Garlock Klozure (PACKING, GASKETS, AND SEALS)

Symbols and Abbreviations

For symbols of chemical elements, see Sec. 6; for abbreviations applying to metric weights and measures and SI units, Sec. 1; SI unit prefixes are listed on p. 1–19. Pairs of parentheses, brackets, etc., are frequently used in this work to indicate corresponding values. For example, the statement that “the cost per kW of a 30,000-kW plant is $86; of a 15,000-kW plant, $98; and of an 8,000-kW plant, $112,” is condensed as follows: The cost per kW of a 30,000 (15,000) [8,000]-kW plant is $86 (98) [112]. In the citation of references readers should always attempt to consult the latest edition of referenced publications. A or Å A AA AAA AAMA AAR AAS ABAI abs a.c. a-c, ac ACI ACM ACRMA ACS ACSR ACV A.D. AEC a-f, af AFBMA AFS AGA AGMA ahp AIChE AIEE AIME AIP AISC AISE AISI Al. Assn. a.m. a-m, am Am. Mach. AMA AMCA amu AN AN-FO ANC

Angstrom unit  1010 m; 3.937  1011 in mass number  N  Z; ampere arithmetical average Am. Automobile Assoc. American Automobile Manufacturers’ Assoc. Assoc. of Am. Railroads Am. Astronautical Soc. Am. Boiler & Affiliated Industries absolute aerodynamic center alternating current Am. Concrete Inst. Assoc. for Computing Machinery Air Conditioning and Refrigerating Manufacturers Assoc. Am. Chemical Soc. aluminum cable steel-reinforced air cushion vehicle anno Domini (in the year of our Lord) Atomic Energy Commission (U.S.) audio frequency Anti-friction Bearings Manufacturers’ Assoc. Am. Foundrymen’s Soc. Am. Gas Assoc. Am. Gear Manufacturers’ Assoc. air horsepower Am. Inst. of Chemical Engineers Am. Inst. of Electrical Engineers (see IEEE) Am. Inst. of Mining Engineers Am. Inst. of Physics American Institute of Steel Construction, Inc. Am. Iron & Steel Engineers Am. Iron and Steel Inst. Aluminum Association ante meridiem (before noon) amplitude modulation Am. Machinist (New York) Acoustical Materials Assoc. Air Moving & Conditioning Assoc., Inc. atomic mass unit ammonium nitrate (explosive); Army-Navy Specification ammonium nitrate-fuel oil (explosive) Army-Navy Civil Aeronautics Committee

ANS ANSI antilog API approx APWA AREA ARI ARS ASCE ASHRAE ASLE ASM ASME ASST ASTM ASTME atm Auto. Ind. avdp avg, ave AWG AWPA AWS AWWA b bar B&S bbl B.C. B.C.C. Bé B.G. bgd BHN bhp BLC B.M. bmep B of M, BuMines

Am. Nuclear Soc. American National Standards Institute antilogarithm of Am. Petroleum Inst. approximately Am. Public Works Assoc. Am. Railroad Eng. Assoc. Air Conditioning and Refrigeration Inst. Am. Rocket Soc. Am. Soc. of Civil Engineers Am. Soc. of Heating, Refrigerating, and Air Conditioning Engineers Am. Soc. of Lubricating Engineers Am. Soc. of Metals Am. Soc. of Mechanical Engineers Am. Soc. of Steel Treating Am. Soc. for Testing and Materials Am. Soc. of Tool & Manufacturing Engineers atmosphere Automotive Industries (New York) avoirdupois average Am. Wire Gage Am. Wood Preservation Assoc. American Welding Soc. American Water Works Assoc. barns barometer Brown & Sharp (gage); Beams and Stringers barrels before Christ body centered cubic Baumé (degrees) Birmingham gage (hoop and sheet) billions of gallons per day Brinnell Hardness Number brake horsepower boundary layer control board measure; bench mark brake mean effective pressure Bureau of Mines

xix

xx

SYMBOLS AND ABBREVIATIONS

BOD bp Bq bsfc BSI Btu Btub, Btu/h bu Bull. Buweaps BWG c C C CAB CAGI cal C-B-R CBS cc, cm3 CCR c to c cd c.f. cf. cfh, ft3/h cfm, ft3/min C.F.R. cfs, ft3/s cg cgs Chm. Eng. chu C.I. cir cir mil cm CME C.N. coef COESA col colog const cos cos1 cosh cosh1 cot cot1 coth coth1 covers c.p. cp cp CP CPH cpm cycles/min cps, cycles/s CSA csc csc1 csch csch1 cu cyl

biochemical oxygen demand boiling point bequerel brake specific fuel consumption British Standards Inst. British thermal units Btu per hr bushels Bulletin Bureau of Weapons, U.S. Navy Birmingham wire gage velocity of light degrees Celsius (centigrade) coulomb Civil Aeronautics Board Compressed Air & Gas Inst. calories chemical, biological & radiological (filters) Columbia Broadcasting System cubic centimetres critical compression ratio center to center candela centrifugal force confer (compare) cubic feet per hour cubic feet per minute Cooperative Fuel Research cubic feet per second center of gravity centimetre-gram-second Chemical Eng’g (New York) centrigrade heat unit cast iron circular circular mils centimetres Chartered Mech. Engr. (IMechE) cetane number coefficient U.S. Committee on Extension to the Standard Atmosphere column cologarithm of constant cosine of angle whose cosine is, inverse cosine of hyperbolic cosine of inverse hyperbolic cosine of cotangent of angle whose cotangent is (see cos1) hyperbolic cotangent of inverse hyperbolic cotangent of coversed sine of circular pitch; center of pressure candle power coef of performance chemically pure close packed hexagonal cycles per minute cycles per second Canadian Standards Assoc. cosecant of angle whose cosecant is (see cos1) hyperbolic cosecant of inverse hyperbolic cosecant of cubic cylinder

db, dB d-c, dc def deg diam. (dia) DO D2O d.p. DP DPH DST d 2 tons DX e EAP EDR EEI eff e.g. ehp EHV El. Wld. elec elong emf Engg. Engr. ENT EP ERDA Eq. est etc. et seq. eV evap exp exsec ext F F FAA F.C. FCC F.C.C. ff. fhp Fig. F.I.T. f-m, fm F.O.B. FP FPC fpm, ft/min fps ft/s F.S. FSB fsp ft fc fL ft  lb g g gal

decibel direct current definition degrees diameter dissolved oxygen deuterium (heavy water) double pole Diametral pitch diamond pyramid hardness daylight saving time breaking strength, d  chain wire diam. in. direct expansion base of Napierian logarithmic system ( 2.7182) equivalent air pressure equivalent direct radiation Edison Electric Inst. efficiency exempli gratia (for example) effective horsepower extra high voltage Electrical World (New York) electric elongation electromotive force Engineering (London) The Engineer (London) emergency negative thrust extreme pressure (lubricant) Energy Research & Development Administration (successor to AEC; see also NRC) equation estimated et cetera (and so forth) et sequens (and the following) electron volts evaporation exponential function of exterior secant of external degrees Fahrenheit farad Federal Aviation Agency fixed carbon, % Federal Communications Commission; Federal Constructive Council face-centered-cubic (alloys) following (pages) friction horsepower figure Federal income tax frequency modulation free on board (cars) fore perpendicular Federal Power Commission feet per minute foot-pound-second system feet per second Federal Specifications Federal Specifications Board fiber saturation point feet foot candles foot lamberts foot-pounds acceleration due to gravity grams gallons

SYMBOLS AND ABBREVIATIONS gc GCA g  cal gd G.E. GEM GFI G.M. GMT GNP gpcd gpd gpm, gal/min gps, gal/s gpt H h h HEPA h-f, hf hhv horiz hp h-p HPAC hp  hr hr, h HSS H.T. HTHW Hz IACS IAeS ibid. ICAO ICC ICE ICI I.C.T. I.D., ID i.e. IEC IEEE IES i-f, if IGT ihp IMechE imep Imp in., in in  lb, in  lb INA Ind. & Eng. Chem. int i-p, ip ipm, in/min ipr IPS IRE IRS ISO isoth ISTM IUPAC

gigacycles per second ground-controlled approach gram-calories Gudermannian of General Electric Co. ground effect machine gullet feed index General Motors Co. Greenwich Mean Time gross national product gallons per capita day gallons per day, grams per denier gallons per minute gallons per second grams per tex henry Planck’s constant  6.624  1027 org-sec Planck’s constant, h  h/2 high efficiency particulate matter high frequency high heat value horizontal horsepower high-pressure Heating, Piping, & Air Conditioning (Chicago) horsepower-hour hours high speed steel heat-treated high temperature hot water hertz  1 cycle/s (cps) International Annealed Copper Standard Institute of Aerospace Sciences ibidem (in the same place) International Civil Aviation Organization Interstate Commerce Commission Inst. of Civil Engineers International Commission on Illumination International Critical Tables inside diameter id est (that is) International Electrotechnical Commission Inst. of Electrical & Electronics Engineers (successor to AIEE, q.v.) Illuminating Engineering Soc. intermediate frequency Inst. of Gas Technology indicated horsepower Inst. of Mechanical Engineers indicated mean effective pressure Imperial inches inch-pounds Inst. of Naval Architects Industrial & Eng’g Chemistry (Easton, PA) internal intermediate pressure inches per minute inches per revolution iron pipe size Inst. of Radio Engineers (see IEEE) Internal Revenue Service International Organization for Standardization isothermal International Soc. for Testing Materials International Union of Pure & Applied Chemistry

J J&P Jour. JP k K K kB kc kcps kg kg  cal kg  m kip kips km kmc kmcps kpsi ksi kts kVA kW kWh L l, L £ lb L.B.P. lhv lim lin ln loc. cit. log LOX l-p, lp LPG lpw, lm/W lx L.W.L. lm m M mA Machy. max MBh mc m.c. Mcf mcps Mech. Eng. mep METO me V MF mhc mi MIL-STD min mip MKS MKSA mL ml, mL mlhc mm

joule joists and planks Journal jet propulsion fuel isentropic exponent; conductivity degrees Kelvin (Celsius abs) Knudsen number kilo Btu (1000 Btu) kilocycles kilocycles per second kilograms kilogram-calories kilogram-metres 1000 lb or 1 kilo-pound thousands of pounds kilometres kilomegacycles per second kilomegacycles per second thousands of pounds per sq in one kip per sq in, 1000 psi (lb/in2) knots kilovolt-amperes kilowatts kilowatt-hours lamberts litres Laplace operational symbol pounds length between perpendiculars low heat value limit linear Napierian logarithm of loco citato (place already cited) common logarithm of liquid oxygen explosive low pressure liquified petroleum gas lumens per watt lux load water line lumen metres thousand; Mach number; moisture, % milliamperes Machinery (New York) maximum thousands of Btu per hr megacycles per second moisture content thousand cubic feet megacycles per second Mechanical Eng’g (ASME) mean effective pressure maximum, except during take-off million electron volts maintenance factor mean horizontal candles mile U.S. Military Standard minutes; minimum mean indicated pressure metre-kilogram-second system metre-kilogram-second-ampere system millilamberts millilitre  1.000027 cm3 mean lower hemispherical candles millimetres

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xxii

SYMBOLS AND ABBREVIATIONS

mm-free mmf mol mp MPC mph, mi/h MRT ms msc MSS mu MW MW day MWT n N N Ns NA NAA NACA NACM NASA nat. NBC NBFU NBS NCN NDHA NEC®

NEMA NFPA NIST NLGI nm No. (Nos.) NPSH NRC NTP O.D., OD O.H. O.N. op. cit. OSHA OSW OTS oz p. (pp.) Pa P.C. PE PEG P.E.L. PETN pf PFI PIV p.m. PM P.N. ppb PPI ppm press

mineral matter free magnetomotive force mole melting point maximum permissible concentration miles per hour mean radiant temperature manuscript; milliseconds mean spherical candles Manufacturers Standardization Soc. of the Valve & Fittings Industry micron, micro megawatts megawatt day mean water temperature polytropic exponent number (in mathematical tables) number of neutrons; newton specific speed not available National Assoc. of Accountants National Advisory Committee on Aeronautics (see NASA) National Assoc. of Chain Manufacturers National Aeronautics and Space Administration natural National Broadcasting Company National Board of Fire Underwriters National Bureau of Standards (see NIST) nitrocarbonitrate (explosive) National District Hearing Assoc. National Electric Code® (National Electrical Code® and NEC® are registered trademarks of the National Fire Protection Association, Inc., Quincy, MA.) National Electrical Manufacturers Assoc. National Fire Protection Assoc. National Institute of Standards and Technology National Lubricating Grease Institute nautical miles number(s) net positive suction head Nuclear Regulator Commission (successor to AEC; see also ERDA) normal temperature and pressure outside diameter (pipes) open-hearth (steel) octane number opere citato (work already cited) Occupational Safety & Health Administration Office of Saline Water Office of Technical Services, U.S. Dept. of Commerce ounces page (pages) pascal propulsive coefficient polyethylene polyethylene glycol proportional elastic limit an explosive power factor Pipe Fabrication Inst. peak inverse voltage post meridiem (after noon) preventive maintenance performance number parts per billion plan position indicator parts per million pressure

Proc. PSD psi, lb/in2 psia psig pt PVC Q qt q.v. r R R rad RBE R-C RCA R&D RDX rem rev r-f, rf RMA rms rpm, r/min rps, r/s RSHF ry. s s S SAE sat SBI scfm SCR sec sec–1 Sec. sech sech–1 segm SE No. SEI sfc sfm, sfpm shp SI sin sin–1 sinh sinh–1 SME SNAME SP sp specif sp gr sp ht spp SPS sq sr SSF SSU std

Proceedings power spectral density, g2/cps lb per sq in lb per sq in. abs lb per sq in. gage point; pint polyvinyl chloride 1018 Btu quarts quod vide (which see) roentgens gas constant deg Rankine (Fahrenheit abs); Reynolds number radius; radiation absorbed dose; radian see rem resistor-capacitor Radio Corporation of America research and development cyclonite, a military explosive Roentgen equivalent man (formerly RBE) revolutions radio frequency Rubber Manufacturers Assoc. square root of mean square revolutions per minute revolutions per second room sensible heat factor railway entropy seconds sulfur, %; siemens Soc. of Automotive Engineers saturated steel Boiler Inst. standard cu ft per min silicon controlled rectifier secant of angle whose secant is (see cos–1) Section hyperbolic secant of inverse hyperbolic secant of segment steam emulsion number Structural Engineering Institute specific fuel consumption, lb per hphr surface feet per minute shaft horsepower International System of Units (Le Systéme International d’Unites) sine of angle whose sine is (see cos–1) hyperbolic sine of inverse hyperbolic sine of Society of Manufacturing Engineers (successor to ASTME) Soc. of Naval Architects and Marine Engineers static pressure specific specification specific gravity specific heat species unspecified (botanical) standard pipe size square steradian sec Saybolt Furol seconds Saybolt Universal (same as SUS) standard

SYMBOLS AND ABBREVIATIONS SUS SWG T TAC tan tan1 tanh tanh1 TDH TEL temp THI thp TNT torr TP tph tpi TR Trans. T.S. tsi ttd UHF UKAEA UL ult UMS USAF USCG USCS USDA USFPL USGS USHEW USN USP

Saybolt Universal seconds (same as SSU) Standard (British) wire gage tesla Technical Advisory Committee on Weather Design Conditions (ASHRAE) tangent of angle whose tangent is (see cos1) hyperbolic tangent of inverse hyperbolic tangent of total dynamic head tetraethyl lead temperature temperature-humidity (discomfort) index thrust horsepower trinitrotoluol (explosive)  1 mm Hg  1.332 millibars (1/760) atm  (1.013250/760) dynes per cm2 total pressure tons per hour turns per in transmitter-receiver Transactions tensile strength; tensile stress tons per sq in terminal temperature difference ultra high frequency United Kingdom Atomic Energy Authority Underwriters’ Laboratory ultimate universal maintenance standards U.S. Air Force U.S. Coast Guard U.S. Commercial Standard; U.S. Customary System U.S. Dept. of Agriculture U.S. Forest Products Laboratory U.S. Geologic Survey U.S. Dept. of Health, Education & Welfare U.S. Navy U.S. pharmacopoeia

USPHS USS USSG UTC V VCF VCI VDI vel vers vert VHF VI viz. V.M. vol VP vs. W Wb W&M w.g. WHO W.I. W.P.A. wt yd Y.P. yr Y.S. z Zeit.  mc s, s m mm

U.S. Public Health Service United States Standard U.S. Standard Gage Coordinated Universal Time volt visual comfort factor visual comfort index Verein Deutscher Ingenieure velocity versed sine of vertical very high frequency viscosity index videlicet (namely) volatile matter, % volume velocity pressure versus watt weber Washburn & Moen wire gage water gage World Health Organization wrought iron Western Pine Assoc. weight yards yield point year(s) yield strength; yield stress atomic number, figure of merit Zeitschrift mass defect microcurie Boltzmann constant micro ( 106 ), as in ms micrometre (micron)  106 m (103 mm) ohm

2 S8  ` 2 3 2 ⬖ y () [] {}

not equal to approaches varies as infinity square root of cube root of therefore parallel to parentheses, brackets and braces; quantities enclosed by them to be taken together in multiplying, dividing, etc. length of line from A to B pi ( 3.14159) degrees minutes seconds angle differential of x (delta) difference increment of x partial derivative of u with respect to x integral of

MATHEMATICAL SIGNS AND SYMBOLS    

( )   / : ⬋

 V W  ; , L >  

plus (sign of addition) positive minus (sign of subtraction) Negative plus or minus (minus or plus) times, by (multiplication sign) multiplied by sign of division divided by ratio sign, divided by, is to equals, as (proportion) less than greater than much less than much greater than equals identical with similar to approximately equals approximately equals, congruent equal to or less than equal to or greater than

AB p  r rr l dx  x 'u/'x e

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xxiv

SYMBOLS AND ABBREVIATIONS

a

3

integral of, between limits a and b

b

r o f (x), F(x) exp x  ex = =2 £

line integral around a closed path (sigma) summation of functions of x [e  2.71828 (base of natural, or Napierian, logarithms)] del or nabla, vector differential operator Laplacian operator Laplace operational symbol

4! |x| # x $ x AB AB

factorial 4  4  3  2  1 absolute value of x first derivative of x with respect to time second derivative of x with respect to time vector product; magnitude of A times magnitude of B times sine of the angle from A to B; AB sin AB scalar product; magnitude of A times magnitude of B times cosine of the angle from A to B; AB cos AB

The Editors

EUGENE A. AVALLONE, Editor, is Professor of Mechanical Engineering, Emeritus, The City College of the City University of New York. He has been engaged for many years as a consultant to industry and to a number of local and national governmental agencies. THEODORE BAUMEISTER, III, Editor, is now retired from Du Pont where he was an internal consultant. His specialties are operations research, business decision making, and longrange planning. He has also taught financial modeling in the United States, South America, and the Far East. ALI M. SADEGH, Editor, is Professor of Mechanical Engineering, The City College of the City University of New York. He is also Director of the Center for Advanced Engineering Design and Development. He is actively engaged in research in the areas of machine design, manufacturing, and biomechanics. He is a Fellow of ASME and SME.

xiii

Preface to the Eleventh Edition

The evolutionary trends underlying modern engineering practice are grounded not only on the tried and true principles and techniques of the past, but also on more recent and current advances. Thus, in the preparation of the eleventh edition of “Marks’,” the Editors have considered the broad enterprise falling under the rubric of “Mechanical Engineering” and have added to and/or amended the contents to include subject areas that will be of maximum utility to the practicing engineer. That said, the Editors note that the publication of this eleventh edition has been accomplished through the combined and coordinated efforts of contributors, readers, and the Editors. First, we recognize, with pleasure, the input from our many contributors—past, continuing, and those newly engaged. Their contributions have been prepared with care, and are authoritative, informative, and concise. Second, our readers, who are practitioners in their own wise, have found that the global treatment of the subjects presented in the “Marks’” permits of great utility and serves as a convenient ready reference. The reading public has had access to “Marks’” since 1916, when Lionel S. Marks edited the first edition. This eleventh edition follows 90 years later. During the intervening years, “Marks’” and “Handbook for Mechanical Engineers” have become synonymous to a wide readership which includes mechanical engineers, engineers in the associated disciplines, and others. Our readership derives from a wide spectrum of interests, and it appears many find the “Marks’” useful as they pursue their professional endeavors. The Editors consider it a given that every successive edition must balance the requests to broaden the range or depth of subject matter printed, the incorporation of new material which will be useful to the widest possible audience, and the requirement to keep the size of the Handbook reasonable and manageable. We are aware that the current engineering practitioner learns quickly that the revolutionary developments of the recent past soon become standard practice. By the same token, it is prudent to realize that as a consequence of rapid developments, some cutting-edge technologies prove to have a short shelf life and soon are regarded as obsolescent—if not obsolete. The Editors are fortunate to have had, from time to time, input from readers and reviewers, who have proffered cogent commentary and suggestions; a number are included in this edition. Indeed, the synergy between Editors, contributors, and readers has been instrumental in the continuing usefulness of successive editions of “Marks’ Standard Handbook for Mechanical Engineers.” The reader will note that a considerable portion of the tabular data and running text continue to be presented in dual units; i.e., USCS and SI. The date for a projected full transition to SI units is not yet firm, and the “Marks’” reflects that. We look to the future in that regard. Society is in an era of information technology, as manifest by the practicing engineer’s working tools. For example: the ubiquitous personal computer, its derivative use of software programs of a vast variety and number, printers, computer-aided design and drawing, universal access to the Internet, and so on. It is recognized, too, that the great leaps forward which

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PREFACE TO THE ELEVENTH EDITION

are thereby enhanced still require the engineer to exercise sound and rational judgment as to the reliability of the solutions provided. Last, the Handbook is ultimately the responsibility of the Editors. The utmost care has been exercised to avoid errors, but if any inadvertently are included, the Publisher and Editors will appreciate being so informed. Corrections will be incorporated into subsequent printings. Ardsley, NY Newark, DE Franklin Lakes, NJ

EUGENE A. AVALLONE THEODORE BAUMEISTER III ALI M. SADEGH

Contents

For the detailed contents of any section consult the title page of that section.

Contributors ix The Editors xiii Preface to the Eleventh Edition xv Preface to the First Edition xvii Symbols and Abbreviations xix

1. Mathematical Tables and Measuring Units . . . . . . . . . . . . . . 1.1 1.2

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Mathematical Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3. Mechanics of Solids and Fluids . . . . . . . . . . . . . . . . . . . . . .

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4. Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6. Materials of Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5-2 5-14 5-51 5-57 5-63 5-71

General Properties of Materials . . . . . . . . . . . . . . . Iron and Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iron and Steel Castings . . . . . . . . . . . . . . . . . . . . . . Nonferrous Metals and Alloys; Metallic Specialties Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paints and Protective Coatings . . . . . . . . . . . . . . . . Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Materials . . . . . . . . . . . . . . . . . . . . . . . Cement, Mortar, and Concrete . . . . . . . . . . . . . . . . .

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5. Strength of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mechanical Properties of Materials . . . . . Mechanics of Materials . . . . . . . . . . . . . . Pipeline Flexure Stresses . . . . . . . . . . . . Nondestructive Testing . . . . . . . . . . . . . . Experimental Stress and Strain Analysis Mechanics of Composite Materials . . . . .

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vi

CONTENTS

6.10 6.11 6.12 6.13

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6-171 6-180 6-189 6-206

7. Fuels and Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.1 7.2 7.3 7.4 7.5

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Industrial Plants . . . . . . . . . . . . . . . . . . . . . . . . Structural Design of Buildings . . . . . . . . . . . . . Reinforced Concrete Design and Construction Air Conditioning, Heating, and Ventilating . . . . Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound, Noise, and Ultrasonics . . . . . . . . . . . . .

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11. Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10-2 10-4 10-22 10-26 10-42 10-63 10-69

Automotive Engineering . . . . . . . . . . . Railway Engineering . . . . . . . . . . . . . . Marine Engineering . . . . . . . . . . . . . . . Aeronautics . . . . . . . . . . . . . . . . . . . . . Jet Propulsion and Aircraft Propellers Astronautics . . . . . . . . . . . . . . . . . . . . Pipeline Transmission . . . . . . . . . . . . . Containerization . . . . . . . . . . . . . . . . .

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11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

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10-1

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10. Materials Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9-3 9-29 9-56 9-58 9-78 9-93 9-127 9-138 9-154

Materials Holding, Feeding, and Metering Lifting, Hoisting, and Elevating . . . . . . . . . Dragging, Pulling, and Pushing . . . . . . . . Loading, Carrying, and Excavating . . . . . . Conveyor Moving and Handling . . . . . . . . Automatic Guided Vehicles and Robots . . Material Storage and Warehousing . . . . . .

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10.1 10.2 10.3 10.4 10.5 10.6 10.7

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9-1

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9. Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8-3 8-10 8-83 8-111 8-127 8-133 8-138

Sources of Energy . . . . . . . . . Steam Boilers . . . . . . . . . . . . . Steam Engines . . . . . . . . . . . . Steam Turbines . . . . . . . . . . . Power-Plant Heat Exchangers Internal-Combustion Engines Gas Turbines . . . . . . . . . . . . . Nuclear Power . . . . . . . . . . . . Hydraulic Turbines . . . . . . . . .

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9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

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8-1

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8. Machine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7-2 7-30 7-43 7-48 7-54

Mechanism . . . . . . . . . . . . . . . . Machine Elements . . . . . . . . . . Gearing . . . . . . . . . . . . . . . . . . Fluid-Film Bearings . . . . . . . . . Bearings with Rolling Contact Packings, Gaskets, and Seals . Pipe, Pipe Fittings, and Valves

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8.1 8.2 8.3 8.4 8.5 8.6 8.7

Fuels . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbonization of Coal and Gas Making Combustion Furnaces . . . . . . . . . . . . . . Municipal Waste Combustion . . . . . . . . Electric Furnaces and Ovens . . . . . . . .

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Foundry Practice and Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-3

CONTENTS

13.2 13.3 13.4 13.5 13.6 13.7

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13-9 13-29 13-50 13-72 13-77 13-80

14. Fans, Pumps, and Compressors . . . . . . . . . . . . . . . . . . . . . .

14-1

14.1 14.2 14.3 14.4 14.5

Plastic Working of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welding and Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Machining Processes and Machine Tools . . . . . . . . . . . . . . . . . Surface Texture Designation, Production, and Quality Control Woodcutting Tools and Machines . . . . . . . . . . . . . . . . . . . . . . . Precision Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14-2 14-15 14-26 14-39 14-46

15. Electrical and Electronics Engineering . . . . . . . . . . . . . . . .

15-1

15.1 15.2

Displacement Pumps Centrifugal Pumps . . . Compressors . . . . . . . High-Vacuum Pumps . Fans . . . . . . . . . . . . . .

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Electrical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15-2 15-68

16. Instruments and Controls . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-1

16.1 16.2 16.3

Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surveying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-2 16-21 16-52

17. Industrial Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17-1

17.1 17.2 17.3 17.4 17.5 17.6 17.7

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17-3 17-11 17-18 17-25 17-31 17-39 17-43

18. The Regulatory Environment . . . . . . . . . . . . . . . . . . . . . . . .

18-1

18.1 18.2 18.3 18.4

Operations Management . . . . . . . . . . . . . . . Cost Accounting . . . . . . . . . . . . . . . . . . . . . Engineering Statistics and Quality Control Methods Engineering . . . . . . . . . . . . . . . . . Cost of Electric Power . . . . . . . . . . . . . . . . . Human Factors and Ergonomics . . . . . . . . Automatic Manufacturing . . . . . . . . . . . . . .

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18-2 18-18 18-22 18-27

19. Refrigeration, Cryogenics, and Optics . . . . . . . . . . . . . . . . .

19-1

19.1 19.2 19.3

Environmental Control . . . . . . . . . . . Occupational Safety and Health . . . . Fire Protection . . . . . . . . . . . . . . . . . . Patents, Trademarks, and Copyrights

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Mechanical Refrigeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19-2 19-26 19-41

20. Emerging Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20-1

20.1 An Introduction to Microelectromechanical Systems (MEMS) 20.2 Introduction to Nanotechnology . . . . . . . . . . . . . . . . . . . . . . . 20.3 Ferroelectrics/Piezoelectrics and Shape Memory Alloys . . . . 20.4 Introduction to the Finite-Element Method . . . . . . . . . . . . . . . 20.5 Computer-Aided Design, Computer-Aided Engineering, and Variational Design . . . . . . . . . . . . . . . . . . 20.6 Introduction to Computational Fluid Dynamics . . . . . . . . . . . 20.7 Experimental Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Introduction to Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . 20.9 Human Injury Tolerance and Anthropometric Test Devices . . 20.10 Air-Inflated Fabric Structures . . . . . . . . . . . . . . . . . . . . . . . . . 20.11 Robotics, Mechatronics, and Intelligent Automation . . . . . . . 20.12 Rapid Prototyping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.13 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index follows Section 20

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20-3 20-13 20-20 20-28

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20-44 20-51 20-63 20-79 20-104 20-108 20-118 20-132 20-135

vii

Section

1

Mathematical Tables and Measuring Units BY

GEORGE F. BAUMEISTER President, EMC Process Co., Newport, DE JOHN T. BAUMEISTER Manager, Product Compliance Test Center, Unisys Corp.

1.1 MATHEMATICAL TABLES by George F. Baumeister Segments of Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Compound Interest and Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Decimal Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 1.2 MEASURING UNITS by John T. Baumeister U.S. Customary System (USCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 Metric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17

The International System of Units (SI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17 Systems of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Terrestrial Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Mohs Scale of Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Density and Relative Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26 Conversion and Equivalency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-27

1.1 MATHEMATICAL TABLES by George F. Baumeister REFERENCES FOR MATHEMATICAL TABLES: Dwight, “Mathematical Tables of Elementary and Some Higher Mathematical Functions,” McGraw-Hill. Dwight, “Tables of Integrals and Other Mathematical Data,” Macmillan. Jahnke and Emde, “Tables of Functions,” B. G. Teubner, Leipzig, or Dover. Pierce-Foster,

“A Short Table of Integrals,” Ginn. “Mathematical Tables from Handbook of Chemistry and Physics,” Chemical Rubber Co. “Handbook of Mathematical Functions,” NBS.

1-1

1-2

MATHEMATICAL TABLES

Table 1.1.1 Segments of Circles, Given h/c Given: h  height; c  chord. To find the diameter of the circle, the length of arc, or the area of the segment, form the ratio h/c, and find from the table the value of (diam /c), (arc/c); then, by a simple multiplication, diam  c  (diam/c) arc  c  (arc/c) area  h  c  (area/h  c) The table gives also the angle subtended at the center, and the ratio of h to D. h c

Diam c

.00 1 2 3 4

25.010 12.520 8.363 6.290

.05 6 7 8 9

5.050 4.227 3.641 3.205 2.868

.10 1 2 3 4

2.600 2.383 2.203 2.053 1.926

.15 6 7 8 9

1.817 1.723 1.641 1.569 1.506

.20 1 2 3 4

1.450 1.400 1.356 1.317 1.282

.25 6 7 8 9

1.250 1.222 1.196 1.173 1.152

.30 1 2 3 4

1.133 1.116 1.101 1.088 1.075

.35 6 7 8 9

1.064 1.054 1.046 1.038 1.031

.40 1 2 3 4

1.025 1.020 1.015 1.011 1.008

.45 6 7 8 9

1.006 1.003 1.002 1.001 1.000

.50

1.000

Diff

12490 *4157 *2073 *1240 *823 *586 *436 *337 *268 *217 *180 *150 *127 *109 *94 *82 *72 *63 56 50 44 39 35 32 28 26 23 21 19 17 15 13 13 11 10 8 8 7 6 5 5 4 3 2 3 1 1 1 0

* Interpolation may be inaccurate at these points.

Arc c 1.000 1.000 1.001 1.002 1.004 1.007 1.010 1.013 1.017 1.021 1.026 1.032 1.038 1.044 1.051 1.059 1.067 1.075 1.084 1.094 1.103 1.114 1.124 1.136 1.147 1.159 1.171 1.184 1.197 1.211 1.225 1.239 1.254 1.269 1.284 1.300 1.316 1.332 1.349 1.366 1.383 1.401 1.419 1.437 1.455 1.474 1.493 1.512 1.531 1.551 1.571

Diff 0 1 1 2 3 3 3 4 4 5 6 6 6 7 8 8 8 9 10 9 11 10 12 11 12 12 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 18 19 19 19 19 20 20

Area h3c .6667 .6667 .6669 .6671 .6675 .6680 .6686 .6693 .6701 .6710 .6720 .6731 .6743 .6756 .6770 .6785 .6801 .6818 .6836 .6855 .6875 .6896 .6918 .6941 .6965 .6989 .7014 .7041 .7068 .7096 .7125 .7154 .7185 .7216 .7248 .7280 .7314 .7348 .7383 .7419 .7455 .7492 .7530 .7568 .7607 .7647 .7687 .7728 .7769 .7811 .7854

Diff 0 2 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 24 25 27 27 28 29 29 31 31 32 32 34 34 35 36 36 37 38 38 39 40 40 41 41 42 43

Central angle, v 0.008 4.58 9.16 13.73 18.30 22.848 27.37 31.88 36.36 40.82 45.248 49.63 53.98 58.30 62.57 66.808 70.98 75.11 79.20 83.23 87.218 91.13 95.00 98.81 102.56 106.268 109.90 113.48 117.00 120.45 123.868 127.20 130.48 133.70 136.86 139.978 143.02 146.01 148.94 151.82 154.648 157.41 160.12 162.78 165.39 167.958 170.46 172.91 175.32 177.69 180.008

Diff 458 458 457 457 454 453 451 448 446 442 439 435 432 427 423 418 413 409 403 399 392 387 381 375 370 364 358 352 345 341 334 328 322 316 311 305 299 293 288 282 277 271 266 261 256 251 245 241 237 231

h Diam .0000 .0004 .0016 .0036 .0064 .0099 .0142 .0192 .0250 .0314 .0385 .0462 .0545 .0633 .0727 .0826 .0929 .1036 .1147 .1263 .1379 .1499 .1622 .1746 .1873 .2000 .2128 .2258 .2387 .2517 .2647 .2777 .2906 .3034 .3162 .3289 .3414 .3538 .3661 .3783 .3902 .4021 .4137 .4252 .4364 .4475 .4584 .4691 .4796 .4899 .5000

Diff 4 12 20 28 35 43 50 58 64 71 77 83 88 94 99 103 107 111 116 116 120 123 124 127 127 128 130 129 130 130 130 129 128 128 127 125 124 123 122 119 119 116 115 112 111 109 107 105 103 101

MATHEMATICAL TABLES

1-3

Table 1.1.2 Segments of Circles, Given h/D Given: h  height; D  diameter of circle. To find the chord, the length of arc, or the area of the segment, form the ratio h/D, and find from the table the value of (chord/D), (arc/D), or (area/D2); then by a simple multiplication, chord  D  (chord/D) arc  D  (arc/D) area  D 2  (area/D 2) This table gives also the angle subtended at the center, the ratio of the arc of the segment of the whole circumference, and the ratio of the area of the segment to the area of the whole circle. h D

Arc D

.00 1 2 3 4

0.000 .2003 .2838 .3482 .4027

.05 6 7 8 9

.4510 .4949 .5355 .5735 .6094

.10 1 2 3 4

.6435 .6761 .7075 .7377 .7670

.15 6 7 8 9

.7954 .8230 .8500 .8763 .9021

.20 1 2 3 4

0.9273 0.9521 0.9764 1.0004 1.0239

.25 6 7 8 9

1.0472 1.0701 1.0928 1.1152 1.1374

.30 1 2 3 4

1.1593 1.1810 1.2025 1.2239 1.2451

.35 6 7 8 9

1.2661 1.2870 1.3078 1.3284 1.3490

.40 1 2 3 4

1.3694 1.3898 1.4101 1.4303 1.4505

.45 6 7 8 9

1.4706 1.4907 1.5108 1.5308 1.5508

.50

1.5708

Diff 2003 *835 *644 *545 *483 *439 *406 *380 *359 *341 *326 *314 *302 *293 *284 276 270 263 258 252 248 243 240 235 233 229 227 224 222 219 217 215 214 212 210 209 208 206 206 204 204 203 202 202 201 201 201 200 200 200

Area D2 .0000 .0013 .0037 .0069 .0105 .0147 .0192 .0242 .0294 .0350 .0409 .0470 .0534 .0600 .0668 .0739 .0811 .0885 .0961 .1039 .1118 .1199 .1281 .1365 .1449 .1535 .1623 .1711 .1800 .1890 .1982 .2074 .2167 .2260 .2355 .2450 .2546 .2642 .2739 .2836 .2934 .3032 .3130 .3229 .3328 .3428 .3527 .3627 .3727 .3827 .3927

* Interpolation may be inaccurate at these points.

Diff 13 24 32 36 42 45 50 52 56 59 61 64 66 68 71 72 74 76 78 79 81 82 84 84 86 88 88 89 90 92 92 93 93 95 95 96 96 97 97 98 98 98 99 99 100 99 100 100 100 100

Central angle, v 0.008 22.96 32.52 39.90 46.15 51.688 56.72 61.37 65.72 69.83 73.748 77.48 81.07 84.54 87.89 91.158 94.31 97.40 100.42 103.37 106.268 109.10 111.89 114.63 117.34 120.008 122.63 125.23 127.79 130.33 132.848 135.33 137.80 140.25 142.67 145.088 147.48 149.86 152.23 154.58 156.938 159.26 161.59 163.90 166.22 168.528 170.82 173.12 175.41 177.71 180.008

Diff 2296 * 956 * 738 * 625 * 553 *

504 465 * 435 * 411 * 391 *

*

374 359 * 347 * 335 * 326 *

316 309 302 295 289 284 279 274 271 266 263 260 256 254 251 249 247 245 242 241 240 238 237 235 235 233 233 231 232 230 230 230 229 230 229

Chord D .0000 .1990 .2800 .3412 .3919 .4359 .4750 .5103 .5426 .5724 .6000 .6258 .6499 .6726 .6940 .7141 .7332 .7513 .7684 .7846 .8000 .8146 .8285 .8417 .8542 .8660 .8773 .8879 .8980 .9075 .9165 .9250 .9330 .9404 .9474 .9539 .9600 .9656 .9708 .9755 .9798 .9837 .9871 .9902 .9928 .9950 .9968 .9982 .9992 .9998 1.0000

Diff *1990 *810 *612 *507 *440 *391 *353 *323 *298 *276 *258 *241 *227 *214 *201 *191 *181 *171 162 154 146 139 132 125 118 113 106 101 95 90 85 80 74 70 65 61 56 52 47 43 39 34 31 26 22 18 14 10 6 2

Arc Circum .0000 .0638 .0903 .1108 .1282 .1436 .1575 .1705 .1826 .1940 .2048 .2152 .2252 .2348 .2441 .2532 .2620 .2706 .2789 .2871 .2952 .3031 .3108 .3184 .3259 .3333 .3406 .3478 .3550 .3620 .3690 .3759 .3828 .3896 .3963 .4030 .4097 .4163 .4229 .4294 .4359 .4424 .4489 .4553 .4617 .4681 .4745 .4809 .4873 .4936 .5000

Diff *638 *265 *205 *174 *154 *139 *130 121 114 108 104 100 96 93 91 88 86 83 82 81 79 77 76 75 74 73 72 72 70 70 69 69 68 67 67 67 66 66 65 65 65 65 64 64 64 64 64 64 63 64

Area Circle .0000 .0017 .0048 .0087 .0134 .0187 .0245 .0308 .0375 .0446 .0520 .0598 .0680 .0764 .0851 .0941 .1033 .1127 .1224 .1323 .1424 .1527 .1631 .1738 .1846 .1955 .2066 .2178 .2292 .2407 .2523 .2640 .2759 .2878 .2998 .3119 .3241 .3364 .3487 .3611 .3735 .3860 .3986 .4112 .4238 .4364 .4491 .4618 .4745 .4873 .5000

Diff 17 31 39 47 53 58 63 67 71 74 78 82 84 87 90 92 94 97 99 101 103 104 107 108 109 111 112 114 115 116 117 119 119 120 121 122 123 123 124 124 125 126 126 126 126 127 127 127 128 127

1-4

MATHEMATICAL TABLES

Table 1.1.3 Regular Polygons n  number of sides v  3608/n  angle subtended at the center by one side v v a  length of one side 5 R a2 sin b 5 r a2 tan b 2 2 v v R  radius of circumscribed circle 5 a a1⁄2 csc b 5 r asec b 2 2 v v r  radius of inscribed circle 5 R acos b 5 a a1⁄2 cot b 2 2 v v Area  a 2 a1⁄4 n cot b 5 R2 s 1⁄2 n sin vd 5 r 2 an tan b 2 2 n

v

3 4 5 6

1208 908 728 608

Area a2

Area R2

Area r2

R a

R r

a R

a r

r R

r a

0.4330 1.000 1.721 2.598

1.299 2.000 2.378 2.598

5.196 4.000 3.633 3.464

0.5774 0.7071 0.8507 1.0000

2.000 1.414 1.236 1.155

1.732 1.414 1.176 1.000

3.464 2.000 1.453 1.155

0.5000 0.7071 0.8090 0.8660

0.2887 0.5000 0.6882 0.8660

3.634 4.828 6.182 7.694

2.736 2.828 2.893 2.939

3.371 3.314 3.276 3.249

1.152 1.307 1.462 1.618

1.110 1.082 1.064 1.052

0.8678 0.7654 0.6840 0.6180

0.9631 0.8284 0.7279 0.6498

0.9010 0.9239 0.9397 0.9511

1.038 1.207 1.374 1.539

7 8 9 10

518.43 458 408 368

12 15 16 20

308 248 228.50 188

11.20 17.64 20.11 31.57

3.000 3.051 3.062 3.090

3.215 3.188 3.183 3.168

1.932 2.405 2.563 3.196

1.035 1.022 1.020 1.013

0.5176 0.4158 0.3902 0.3129

0.5359 0.4251 0.3978 0.3168

0.9659 0.9781 0.9808 0.9877

1.866 2.352 2.514 3.157

24 32 48 64

158 118.25 78.50 58.625

45.58 81.23 183.1 325.7

3.106 3.121 3.133 3.137

3.160 3.152 3.146 3.144

3.831 5.101 7.645 10.19

1.009 1.005 1.002 1.001

0.2611 0.1960 0.1308 0.0981

0.2633 0.1970 0.1311 0.0983

0.9914 0.9952 0.9979 0.9968

3.798 5.077 7.629 10.18

Table 1.1.4 (n)0  1

Binomial Coefficients nsn 2 1d nsn 2 1dsn 2 2d nsn 2 1dsn 2 2d c[n 2 sr 2 1d] n etc. in general sndr 5 snd2 5 snd3 5 . Other notations: nCr 5 a b 5 sndr 132 13233 1 3 2 3 3 3 c3 r r

(n)I  n

n

(n)0

(n)1

(n)2

(n)3

(n)4

(n)5

(n)6

(n)7

(n)8

(n)9

(n)10

(n)11

(n)12

(n)13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 4 10 20 35 56 84 120 165 220 286 364 455

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 5 15 35 70 126 210 330 495 715 1001 1365

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 6 21 56 126 252 462 792 1287 2002 3003

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 7 28 84 210 462 924 1716 3003 5005

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 8 36 120 330 792 1716 3432 6435

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 9 45 165 495 1287 3003 6435

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 10 55 220 715 2002 5005

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 11 66 286 1001 3003

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 12 78 364 1365

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 13 91 455

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ 1 14 105

1 3 6 10 15 21 28 36 45 55 66 78 91 105

NOTE: For n  14, (n)14  1; for n  15, (n)14  15, and (n)15  1.

MATHEMATICAL TABLES

1-5

Table 1.1.5 Compound Interest. Amount of a Given Principal The amount A at the end of n years of a given principal P placed at compound interest today is A  P  x or A  P  y, according as the interest (at the rate of r percent per annum) is compounded annually, or continuously; the factor x or y being taken from the following tables. Values of x (interest compounded annually: A  P  x) Years

r2

3

4

5

6

7

1 2 3 4 5

1.0200 1.0404 1.0612 1.0824 1.1041

1.0300 1.0609 1.0927 1.1255 1.1593

1.0400 1.0816 1.1249 1.1699 1.2167

1.0500 1.1025 1.1576 1.2155 1.2763

1.0600 1.1236 1.1910 1.2625 1.3382

1.0700 1.1449 1.2250 1.3108 1.4026

1.0800 1.1664 1.2597 1.3605 1.4693

1.1000 1.2100 1.3310 1.4641 1.6105

1.1200 1.2544 1.4049 1.5735 1.7623

6 7 8 9 10

1.1262 1.1487 1.1717 1.1951 1.2190

1.1941 1.2299 1.2668 1.3048 1.3439

1.2653 1.3159 1.3686 1.4233 1.4802

1.3401 1.4071 1.4775 1.5513 1.6289

1.4185 1.5036 1.5938 1.6895 1.7908

1.5007 1.6058 1.7182 1.8385 1.9672

1.5869 1.7138 1.8509 1.9990 2.1589

1.7716 1.9487 2.1436 2.3579 2.5937

1.9738 2.2107 2.4760 2.7731 3.1058

11 12 13 14 15

1.2434 1.2682 1.2936 1.3195 1.3459

1.3842 1.4258 1.4685 1.5126 1.5580

1.5395 1.6010 1.6651 1.7317 1.8009

1.7103 1.7959 1.8856 1.9799 2.0789

1.8983 2.0122 2.1329 2.2609 2.3966

2.1049 2.2522 2.4098 2.5785 2.7590

2.3316 2.5182 2.7196 2.9372 3.1722

2.8531 3.1384 3.4523 3.7975 4.1772

3.4785 3.8960 4.3635 4.8871 5.4736

16 17 18 19 20

1.3728 1.4002 1.4282 1.4568 1.4859

1.6047 1.6528 1.7024 1.7535 1.8061

1.8730 1.9479 2.0258 2.1068 2.1911

2.1829 2.2920 2.4066 2.5270 2.6533

2.5404 2.6928 2.8543 3.0256 3.2071

2.9522 3.1588 3.3799 3.6165 3.8697

3.4259 3.7000 3.9960 4.3157 4.6610

4.5950 5.0545 5.5599 6.1159 6.7275

6.1304 6.8660 7.6900 8.6128 9.6463

25 30 40 50 60

1.6406 1.8114 2.2080 2.6916 3.2810

2.0938 2.4273 3.2620 4.3839 5.8916

2.6658 3.2434 4.8010 7.1067 10.520

3.3864 4.3219 7.0400 11.467 18.679

4.2919 5.7435 10.286 18.420 32.988

5.4274 7.6123 14.974 29.457 57.946

6.8485 10.063 21.725 46.902 101.26

8

10

12

10.835 17.449 45.259 117.39 304.48

17.000 29.960 93.051 289.00 897.60

10

12

NOTE: This table is computed from the formula x  [1  (r/100)]n.

Values of y (interest compounded continuously: A  P  y) Years

r2

3

4

5

6

7

1 2 3 4 5

1.0202 1.0408 1.0618 1.0833 1.1052

1.0305 1.0618 1.0942 1.1275 1.1618

1.0408 1.0833 1.1275 1.1735 1.2214

1.0513 1.1052 1.1618 1.2214 1.2840

1.0618 1.1275 1.1972 1.2712 1.3499

1.0725 1.1503 1.2337 1.3231 1.4191

1.0833 1.1735 1.2712 1.3771 1.4918

1.1052 1.2214 1.3499 1.4918 1.6487

1.1275 1.2712 1.4333 1.6161 1.8221

6 7 8 9 10

1.1275 1.1503 1.1735 1.1972 1.2214

1.1972 1.2337 1.2712 1.3100 1.3499

1.2712 1.3231 1.3771 1.4333 1.4918

1.3499 1.4191 1.4918 1.5683 1.6487

1.4333 1.5220 1.6161 1.7160 1.8221

1.5220 1.6323 1.7507 1.8776 2.0138

1.6161 1.7507 1.8965 2.0544 2.2255

1.8221 2.0138 2.2255 2.4596 2.7183

2.0544 2.3164 2.6117 2.9447 3.3201

11 12 13 14 15

1.2461 1.2712 1.2969 1.3231 1.3499

1.3910 1.4333 1.4770 1.5220 1.5683

1.5527 1.6161 1.6820 1.7507 1.8221

1.7333 1.8221 1.9155 2.0138 2.1170

1.9348 2.0544 2.1815 2.3164 2.4596

2.1598 2.3164 2.4843 2.6645 2.8577

2.4109 2.6117 2.8292 3.0649 3.3201

3.0042 3.3201 3.6693 4.0552 4.4817

3.7434 4.2207 4.7588 5.3656 6.0496

16 17 18 19 20

1.3771 1.4049 1.4333 1.4623 1.4918

1.6161 1.6653 1.7160 1.7683 1.8221

1.8965 1.9739 2.0544 2.1383 2.2255

2.2255 2.3396 2.4596 2.5857 2.7183

2.6117 2.7732 2.9447 3.1268 3.3201

3.0649 3.2871 3.5254 3.7810 4.0552

3.5966 3.8962 4.2207 4.5722 4.9530

4.9530 5.4739 6.0496 6.6859 7.3891

6.8210 7.6906 8.6711 9.7767 11.023

25 30 40 50 60

1.6487 1.8221 2.2255 2.7183 3.3201

2.1170 2.4596 3.3201 4.4817 6.0496

2.7183 3.3201 4.9530 7.3891 11.023

3.4903 4.4817 7.3891 12.182 20.086

4.4817 6.0496 11.023 20.086 36.598

5.7546 8.1662 16.445 33.115 66.686

7.3891 11.023 24.533 54.598 121.51

FORMULA: y  e(r/100)  n.

8

12.182 20.086 54.598 148.41 403.43

20.086 36.598 121.51 403.43 1339.4

1-6

MATHEMATICAL TABLES Table 1.1.6 Principal Which Will Amount to a Given Sum The principal P, which, if placed at compound interest today, will amount to a given sum A at the end of n years P  A  xr or P  A  yr, according as the interest (at the rate of r percent per annum) is compounded annually, or continuously; the factor xr or yr being taken from the following tables. Values of xr (interest compounded annually: P  A  xr) Years

r2

3

4

5

6

7

8

10

12

1 2 3 4 5

.98039 .96117 .94232 .92385 .90573

.97087 .94260 .91514 .88849 .86261

.96154 .92456 .88900 .85480 .82193

.95238 .90703 .86384 .82270 .78353

.94340 .89000 .83962 .79209 .74726

.93458 .87344 .81630 .76290 .71299

.92593 .85734 .79383 .73503 .68058

.90909 .82645 .75131 .68301 .62092

.89286 .79719 .71178 .63552 .56743

6 7 8 9 10

.88797 .87056 .85349 .83676 .82035

.83748 .81309 .78941 .76642 .74409

.79031 .75992 .73069 .70259 .67556

.74622 .71068 .67684 .64461 .61391

.70496 .66506 .62741 .59190 .55839

.66634 .62275 .58201 .54393 .50835

.63017 .58349 .54027 .50025 .46319

.56447 .51316 .46651 .42410 .38554

.50663 .45235 .40388 .36061 .32197

11 12 13 14 15

.80426 .78849 .77303 .75788 .74301

.72242 .70138 .68095 .66112 .64186

.64958 .62460 .60057 .57748 .55526

.58468 .55684 .53032 .50507 .48102

.52679 .49697 .46884 .44230 .41727

.47509 .44401 .41496 .38782 .36245

.42888 .39711 .36770 .34046 .31524

.35049 .31863 .28966 .26333 .23939

.28748 .25668 .22917 .20462 .18270

16 17 18 19 20

.72845 .71416 .70016 .68643 .67297

.62317 .60502 .58739 .57029 .55368

.53391 .51337 .49363 .47464 .45639

.45811 .43630 .41552 .39573 .37689

.39365 .37136 .35034 .33051 .31180

.33873 .31657 .29586 .27651 .25842

.29189 .27027 .25025 .23171 .21455

.21763 .19784 .17986 .16351 .14864

.16312 .14564 .13004 .11611 .10367

25 30 40 50 60

.60953 .55207 .45289 .37153 .30478

.47761 .41199 .30656 .22811 .16973

.37512 .30832 .20829 .14071 .09506

.29530 .23138 .14205 .08720 .05354

.23300 .17411 .09722 .05429 .03031

.18425 .13137 .06678 .03395 .01726

.14602 .09938 .04603 .02132 .00988

.09230 .05731 .02209 .00852 .00328

.05882 .03338 .01075 .00346 .00111

FORMULA: xr  [1  (r/100)]n  1/x.

Values of yr (interest compounded continuously: P  A  yr) Years

r2

3

4

5

6

7

8

10

12

1 2 3 4 5

.98020 .96079 .94176 .92312 .90484

.97045 .94176 .91393 .88692 .86071

.96079 .92312 .88692 .85214 .81873

.95123 .90484 .86071 .81873 .77880

.94176 .88692 .83527 .78663 .74082

.93239 .86936 .81058 .75578 .70469

.92312 .85214 .78663 .72615 .67032

.90484 .81873 .74082 .67032 .60653

.88692 .78663 .69768 .61878 .54881

6 7 8 9 10

.88692 .86936 .85214 .83527 .81873

.83527 .81058 .78663 .76338 .74082

.78663 .75578 .72615 .69768 .67032

.74082 .70469 .67032 .63763 .60653

.69768 .65705 .61878 .58275 .54881

.65705 .61263 .57121 .53259 .49659

.61878 .57121 .52729 .48675 .44933

.54881 .49659 .44933 .40657 .36788

.48675 .43171 .38289 .33960 .30119

11 12 13 14 15

.80252 .78663 .77105 .75578 .74082

.71892 .69768 .67706 .65705 .63763

.64404 .61878 .59452 .57121 .54881

.57695 .54881 .52205 .49659 .47237

.51685 .48675 .45841 .43171 .40657

.46301 .43171 .40252 .37531 .34994

.41478 .38289 .35345. .32628 .30119

.33287 .30119 .27253 .24660 .22313

.26714 .23693 .21014 .18637 .16530

16 17 18 19 20

.72615 .71177 .69768 .68386 .67032

.61878 .60050 .58275 .56553 .54881

.52729 .50662 .48675 .46767 .44933

.44933 .42741 .40657 .38674 .36788

.38289 .36059 .33960 .31982 .30119

.32628 .30422 .28365 .26448 .24660

.27804 .25666 .23693 .21871 .20190

.20190 .18268 .16530 .14957 .13534

.14661 .13003 .11533 .10228 .09072

25 30 40 50 60

.60653 .54881 .44933 .36788 .30119

.47237 .40657 .30119 .22313 .16530

.36788 .30119 .20190 .13534 .09072

.28650 .22313 .13534 .08208 .04979

.22313 .16530 .09072 .04979 .02732

.17377 .12246 .06081 .03020 .01500

.13534 .09072 .04076 .01832 .00823

.08208 .04979 .01832 .00674 .00248

.04979 .02732 .00823 .00248 .00075

FORMULA: yr  e(r/100)n  1/y.

MATHEMATICAL TABLES

1-7

Table 1.1.7 Amount of an Annuity The amount S accumulated at the end of n years by a given annual payment Y set aside at the end of each year is S  Y  v, where the factor v is to be taken from the following table (interest at r percent per annum, compounded annually). Values of v Years

r2

3

4

5

10

12

1 2 3 4 5

1.0000 2.0200 3.0604 4.1216 5.2040

1.0000 2.0300 3.0909 4.1836 5.3091

1.0000 2.0400 3.1216 4.2465 5.4163

1.0000 2.0500 3.1525 4.3101 5.5256

1.0000 2.0600 3.1836 4.3746 5.6371

6

1.0000 2.0700 3.2149 4.4399 5.7507

7

1.0000 2.0800 3.2464 4.5061 5.8666

8

1.0000 2.1000 3.3100 4.6410 6.1051

1.0000 2.1200 3.3744 4.7793 6.3528

6 7 8 9 10

6.3081 7.4343 8.5830 9.7546 10.950

6.4684 7.6625 8.8923 10.159 11.464

6.6330 7.8983 9.2142 10.583 12.006

6.8019 8.1420 9.5491 11.027 12.578

6.9753 8.3938 9.8975 11.491 13.181

7.1533 8.6540 10.260 11.978 13.816

7.3359 8.9228 10.637 12.488 14.487

7.7156 9.4872 11.436 13.579 15.937

8.1152 10.089 12.300 14.776 17.549

11 12 13 14 15

12.169 13.412 14.680 15.974 17.293

12.808 14.192 15.618 17.086 18.599

13.486 15.026 16.627 18.292 20.024

14.207 15.917 17.713 19.599 21.579

14.972 16.870 18.882 21.015 23.276

15.784 17.888 20.141 22.550 25.129

16.645 18.977 21.495 24.215 27.152

18.531 21.384 24.523 27.975 31.772

20.655 24.133 28.029 32.393 37.280

16 17 18 19 20

18.639 20.012 21.412 22.841 24.297

20.157 21.762 23.414 25.117 26.870

21.825 23.698 25.645 27.671 29.778

23.657 25.840 28.132 30.539 33.066

25.673 28.213 30.906 33.760 36.786

27.888 30.840 33.999 37.379 40.995

30.324 33.750 37.450 41.446 45.762

35.950 40.545 45.599 51.159 57.275

42.753 48.884 55.750 63.440 72.052

25 30 40 50 60

32.030 40.568 60.402 84.579 114.05

36.459 47.575 75.401 112.80 163.05

41.646 56.085 95.026 152.67 237.99

47.727 66.439 120.80 209.35 353.58

54.865 79.058 154.76 290.34 533.13

63.249 94.461 199.64 406.53 813.52

73.106 113.28 259.06 573.77 1253.2

98.347 164.49 442.59 1163.9 3034.8

133.33 241.33 767.09 2400.0 7471.6

FORMULA: v {[1  (r/100)]n  1} (r/100)  (x  1) (r/100).

Table 1.1.8 Annuity Which Will Amount to a Given Sum (Sinking Fund) The annual payment Y which, if set aside at the end of each year, will amount with accumulated interest to a given sum S at the end of n years is Y  S  vr, where the factor vr is given below (interest at r percent per annum, compounded annually). Values of vr Years

r2

3

4

5

6

7

8

10

12

1 2 3 4 5

1.0000 .49505 .32675 .24262 .19216

1.0000 .49261 .32353 .23903 .18835

1.0000 .49020 .32035 .23549 .18463

1.0000 .48780 .31721 .23201 .18097

1.0000 .48544 .31411 .22859 .17740

1.0000 .48309 .31105 .22523 .17389

1.0000 .48077 .30803 .22192 .17046

1.0000 .47619 .30211 .21547 .16380

1.0000 .47170 .29635 .20923 .15741

6 7 8 9 10

.15853 .13451 .11651 .10252 .09133

.15460 .13051 .11246 .09843 .08723

.15076 .12661 .10853 .09449 .08329

.14702 .12282 .10472 .09069 .07950

.14336 .11914 .10104 .08702 .07587

.13980 .11555 .09747 .08349 .07238

.13632 .11207 .09401 .08008 .06903

.12961 .10541 .08744 .07364 .06275

.12323 .09912 .08130 .06768 .05698

11 12 13 14 15

.08218 .07456 .06812 .06260 .05783

.07808 .07046 .06403 .05853 .05377

.07415 .06655 .06014 .05467 .04994

.07039 .06283 .05646 .05102 .04634

.06679 .05928 .05296 .04758 .04296

.06336 .05590 .04965 .04434 .03979

.06008 .05270 .04652 .04130 .03683

.05396 .04676 .04078 .03575 .03147

.04842 .04144 .03568 .03087 .02682

16 17 18 19 20

.05365 .04997 .04670 .04378 .04116

.04961 .04595 .04271 .03981 .03722

.04582 .04220 .03899 .03614 .03358

.04227 .03870 .03555 .03275 .03024

.03895 .03544 .03236 .02962 .02718

.03586 .03243 .02941 .02675 .02439

.03298 .02963 .02670 .02413 .02185

.02782 .02466 .02193 .01955 .01746

.02339 .02046 .01794 .01576 .01388

25 30 40 50 60

.03122 .02465 .01656 .01182 .00877

.02743 .02102 .01326 .00887 .00613

.02401 .01783 .01052 .00655 .00420

.02095 .01505 .00828 .00478 .00283

.01823 .01265 .00646 .00344 .00188

.01581 .01059 .00501 .00246 .00123

.01368 .00883 .00386 .00174 .00080

.01017 .00608 .00226 .00086 .00033

.00750 .00414 .00130 .00042 .00013

FORMULA: v  (r/100) {[1  (r/100)]n  1}  1/v.

1-8

MATHEMATICAL TABLES

Table 1.1.9 Present Worth of an Annuity The capital C which, if placed at interest today, will provide for a given annual payment Y for a term of n years before it is exhausted is C  Y  w, where the factor w is given below (interest at r percent per annum, compounded annually). Values of w Years

r2

3

4

5

6

7

8

10

12

1 2 3 4 5

.98039 1.9416 2.8839 3.8077 4.7135

.97087 1.9135 2.8286 3.7171 4.5797

.96154 1.8861 2.7751 3.6299 4.4518

.95238 1.8594 2.7232 3.5460 4.3295

.94340 1.8334 2.6730 3.4651 4.2124

.93458 1.8080 2.6243 3.3872 4.1002

.92593 1.7833 2.5771 3.3121 3.9927

.90909 1.7355 2.4869 3.1699 3.7908

.89286 1.6901 2.4018 3.0373 3.6048

6 7 8 9 10

5.6014 6.4720 7.3255 8.1622 8.9826

5.4172 6.2303 7.0197 7.7861 8.5302

5.2421 6.0021 6.7327 7.4353 8.1109

5.0757 5.7864 6.4632 7.1078 7.7217

4.9173 5.5824 6.2098 6.8017 7.3601

4.7665 5.3893 5.9713 6.5152 7.0236

4.6229 5.2064 5.7466 6.2469 6.7101

4.3553 4.8684 5.3349 5.7590 6.1446

4.1114 4.5638 4.9676 5.3282 5.6502

11 12 13 14 15

9.7868 10.575 11.348 12.106 12.849

9.2526 9.9540 10.635 11.296 11.938

8.7605 9.3851 9.9856 10.563 11.118

8.3064 8.8633 9.3936 9.8986 10.380

7.8869 8.3838 8.8527 9.2950 9.7122

7.4987 7.9427 8.3577 8.7455 9.1079

7.1390 7.5361 7.9038 8.2442 8.5595

6.4951 6.8137 7.1034 7.3667 7.6061

5.9377 6.1944 6.4235 6.6282 6.8109

16 17 18 19 20

13.578 14.292 14.992 15.678 16.351

12.561 13.166 13.754 14.324 14.877

11.652 12.166 12.659 13.134 13.590

10.838 11.274 11.690 12.085 12.462

10.106 10.477 10.828 11.158 11.470

9.4466 9.7632 10.059 10.336 10.594

8.8514 9.1216 9.3719 9.6036 9.8181

7.8237 8.0216 8.2014 8.3649 8.5136

6.9740 7.1196 7.2497 7.3658 7.4694

25 30 40 50 60

19.523 22.396 27.355 31.424 34.761

17.413 19.600 23.115 25.730 27.676

15.622 17.292 19.793 21.482 22.623

14.094 15.372 17.159 18.256 18.929

12.783 13.765 15.046 15.762 16.161

11.654 12.409 13.332 13.801 14.039

9.0770 9.4269 9.7791 9.9148 9.9672

7.8431 8.0552 8.2438 8.3045 8.3240

10.675 11.258 11.925 12.233 12.377

FORMULA: w  {1  [1  (r/100)]n} [r/100]  v/x.

Table 1.1.10 Annuity Provided for by a Given Capital The annual payment Y provided for a term of n years by a given capital C placed at interest today is Y  C  wr (interest at r percent per annum, compounded annually; the fund supposed to be exhausted at the end of the term). Values of wr Years

r2

1 2 3 4 5

1.0200 .51505 .34675 .26262 .21216

6 7 8 9 10

3

4

5

6

7

8

10

12

1.0300 .52261 .35353 .26903 .21835

1.0400 .53020 .36035 .27549 .22463

1.0500 .53780 .36721 .28201 .23097

1.0600 .54544 .37411 .28859 .23740

1.0700 .55309 .38105 .29523 .24389

1.0800 .56077 .38803 .30192 .25046

1.1000 .57619 .40211 .31547 .26380

1.1200 .59170 .41635 .32923 .27741

.17853 .15451 .13651 .12252 .11133

.18460 .16051 .14246 .12843 .11723

.19076 .16661 .14853 .13449 .12329

.19702 .17282 .15472 .14069 .12950

.20336 .17914 .16104 .14702 .13587

.20980 .18555 .16747 .15349 .14238

.21632 .19207 .17401 .16008 .14903

.22961 .20541 .18744 .17364 .16275

.24323 .21912 .20130 .18768 .17698

11 12 13 14 15

.10218 .09456 .08812 .08260 .07783

.10808 .10046 .09403 .08853 .08377

.11415 .10655 .10014 .09467 .08994

.12039 .11283 .10646 .10102 .09634

.12679 .11928 .11296 .10758 .10296

.13336 .12590 .11965 .11434 .10979

.14008 .13270 .12652 .12130 .11683

.15396 .14676 .14078 .13575 .13147

.16842 .16144 .15568 .15087 .14682

16 17 13 19 20

.07365 .06997 .06670 .06378 .06116

.07961 .07595 .07271 .06981 .06722

.08582 .08220 .07899 .07614 .07358

.09227 .08870 .08555 .08275 .08024

.09895 .09544 .09236 .08962 .08718

.10586 .10243 .09941 .09675 .09439

.11298 .10963 .10670 .10413 .10185

.12782 .12466 .12193 .11955 .11746

.14339 .14046 .13794 .13576 .13388

25 30 40 50 60

.05122 .04465 .03656 .03182 .02877

.05743 .05102 .04326 .03887 .03613

.06401 .05783 .05052 .04655 .04420

.07095 .06505 .05828 .05478 .05283

.07823 .07265 .06646 .06344 .06188

.08581 .08059 .07501 .07246 .07123

.09368 .08883 .08386 .08174 .08080

.11017 .10608 .10226 .10086 .10033

.12750 .12414 .12130 .12042 .12013

FORMULA: wr  [r/100] {1  [1  (r/100)]n}  1/w  v  (r/100).

MATHEMATICAL TABLES Table 1.1.11 Ordinates of the Normal Density Function 1 2 fsxd 5 e2x >2 !2p x

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.0 .1 .2 .3 .4

.3989 .3970 .3910 .3814 .3683

.3989 .3965 .3902 .3802 .3668

.3989 .3961 .3894 .3790 .3653

.3988 .3956 .3885 .3778 .3637

.3986 .3951 .3876 .3765 .3621

.3984 .3945 .3867 .3752 .3605

.3982 .3939 .3857 .3739 .3589

.3980 .3932 .3847 .3725 .3572

.3977 .3925 .3836 .3712 .3555

.3973 .3918 .3825 .3697 .3538

.5 .6 .7 .8 .9

.3521 .3332 .3123 .2897 .2661

.3503 .3312 .3101 .2874 .2637

.3485 .3292 .3079 .2850 .2613

.3467 .3271 .3056 .2827 .2589

.3448 .3251 .3034 .2803 .2565

.3429 .3230 .3011 .2780 .2541

.3410 .3209 .2989 .2756 .2516

.3391 .3187 .2966 .2732 .2492

.3372 .3166 .2943 .2709 .2468

.3352 .3144 .2920 .2685 .2444

1.0 1.1 1.2 1.3 1.4

.2420 .2179 .1942 .1714 .1497

.2396 .2155 .1919 .1691 .1476

.2371 .2131 .1895 .1669 .1456

.2347 .2107 .1872 .1647 .1435

.2323 .2083 .1849 .1626 .1415

.2299 .2059 .1826 .1604 .1394

.2275 .2036 .1804 .1582 .1374

.2251 .2012 .1781 .1561 .1354

.2227 .1989 .1758 .1539 .1334

.2203 .1965 .1736 .1518 .1315

1.5 1.6 1.7 1.8 1.9

.1295 .1109 .0940 .0790 .0656

.1276 .1092 .0925 .0775 .0644

.1257 .1074 .0909 .0761 .0632

.1238 .1057 .0893 .0748 .0620

.1219 .1040 .0878 .0734 .0608

.1200 .1023 .0863 .0721 .0596

.1182 .1006 .0848 .0707 .0584

.1163 .0989 .0833 .0694 .0573

.1154 .0973 .0818 .0681 .0562

.1127 .0957 .0804 .0669 .0551

2.0 2.1 2.2 2.3 2.4

.0540 .0440 .0355 .0283 .0224

.0529 .0431 .0347 .0277 .0219

.0519 .0422 .0339 .0270 .0213

.0508 .0413 .0332 .0264 .0208

.0498 .0404 .0325 .0258 .0203

.0488 .0396 .0317 .0252 .0198

.0478 .0387 .0310 .0246 .0194

.0468 .0379 .0303 .0241 .0189

.0459 .0371 .0297 .0235 .0184

.0449 .0363 .0290 .0229 .0180

2.5 2.6 2.7 2.8 2.9

.0175 .0136 .0104 .0079 .0060

.0171 .0132 .0101 .0077 .0058

.0167 .0129 .0099 .0075 .0056

.0163 .0126 .0096 .0073 .0055

.0158 .0122 .0093 .0071 .0053

.0154 .0119 .0091 .0069 .0051

.0151 .0116 .0088 .0067 .0050

.0147 .0113 .0086 .0065 .0048

.0143 .0110 .0084 .0063 .0047

.0139 .0107 .0081 .0061 .0046

3.0 3.1 3.2 3.3 3.4

.0044 .0033 .0024 .0017 .0012

.0043 .0032 .0023 .0017 .0012

.0042 .0031 .0022 .0016 .0012

.0040 .0030 .0022 .0016 .0011

.0039 .0029 .0021 .0015 .0011

.0038 .0028 .0020 .0015 .0010

.0037 .0027 .0020 .0014 .0010

.0036 .0026 .0019 .0014 .0010

.0035 .0025 .0018 .0013 .0009

.0034 .0025 .0018 .0013 .0009

3.5 3.6 3.7 3.8 3.9

.0009 .0006 .0004 .0003 .0002

.0008 .0006 .0004 .0003 .0002

.0008 .0006 .0004 .0003 .0002

.0008 .0005 .0004 .0003 .0002

.0008 .0005 .0004 .0003 .0002

.0007 .0005 .0004 .0002 .0002

.0007 .0005 .0003 .0002 .0002

.0007 .0005 .0003 .0002 .0002

.0007 .0005 .0003 .0002 .0001

.0006 .0004 .0003 .0002 .0001

NOTE: x is the value in left-hand column  the value in top row. f(x) is the value in the body of the table. Example: x  2.14; f (x)  0.0404.

1-9

1-10

MATHEMATICAL TABLES Table 1.1.12

Cumulative Normal Distribution 1 2 e2t / 2 dt 2` !2p x

Fsxd 5 3 x

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.0 .1 .2 .3 .4

.5000 .5398 .5793 .6179 .6554

.5040 .5438 .5832 .6217 .6591

.5080 .5478 .5871 .6255 .6628

.5120 .5517 .5910 .6293 .6664

.5160 .5557 .5948 .6331 .6700

.5199 .5596 .5987 .6368 .6736

.5239 .5636 .6026 .6406 .6772

.5279 .5675 .6064 .6443 .6808

.5319 .5714 .6103 .6480 .6844

.5359 .5735 .6141 .6517 .6879

.5 .6 .7 .8 .9

.6915 .7257 .7580 .7881 .8159

.6950 .7291 .7611 .7910 .8186

.6985 .7324 .7642 .7939 .8212

.7019 .7357 .7673 .7967 .8238

.7054 .7389 .7703 .7995 .8264

.7088 .7422 .7734 .8023 .8289

.7123 .7454 .7764 .8051 .8315

.7157 .7486 .7793 .8078 .8340

.7190 .7517 .7823 .8106 .8365

.7224 .7549 .7852 .8133 .8389

1.0 1.1 1.2 1.3 1.4

.8413 .8643 .8849 .9032 .9192

.8438 .8665 .8869 .9049 .9207

.8461 .8686 .8888 .9066 .9222

.8485 .8708 .8906 .9082 .9236

.8508 .8729 .8925 .9099 .9251

.8531 .8749 .8943 .9115 .9265

.8554 .8770 .8962 .9131 .9279

.8577 .8790 .8980 .9147 .9292

.8599 .8810 .8997 .9162 .9306

.8621 .8830 .9015 .9177 .9319

1.5 1.6 1.7 1.8 1.9

.9332 .9452 .9554 .9641 .9713

.9345 .9463 .9564 .9649 .9719

.9357 .9474 .9573 .9656 .9726

.9370 .9484 .9582 .9664 .9732

.9382 .9495 .9591 .9671 .9738

.9394 .9505 .9599 .9678 .9744

.9406 .9515 .9608 .9686 .9750

.9418 .9525 .9616 .9693 .9756

.9429 .9535 .9625 .9699 .9761

.9441 .9545 .9633 .9706 .9767

2.0 2.1 2.2 2.3 2.4

.9772 .9812 .9861 .9893 .9918

.9778 .9826 .9864 .9896 .9920

.9783 .9830 .9868 .9898 .9922

.9788 .9834 .9871 .9901 .9925

.9793 .9838 .9875 .9904 .9927

.9798 .9842 .9878 .9906 .9929

.9803 .9846 .9881 .9909 .9931

.9808 .9850 .9884 .9911 .9932

.9812 .9854 .9887 .9913 .9934

.9817 .9857 .9890 .9916 .9936

2.5 2.6 2.7 2.8 2.9

.9938 .9953 .9965 .9974 .9981

.9940 .9955 .9966 .9975 .9982

.9941 .9956 .9967 .9976 .9982

.9943 .9957 .9968 .9977 .9983

.9945 .9959 .9969 .9977 .9984

.9946 .9960 .9970 .9978 .9984

.9948 .9961 .9971 .9979 .9985

.9949 .9962 .9972 .9979 .9985

.9951 .9963 .9973 .9980 .9986

.9952 .9964 .9974 .9981 .9986

3.0 3.1 3.2 3.3 3.4

.9986 .9990 .9993 .9995 .9997

.9987 .9991 .9993 .9995 .9997

.9987 .9991 .9994 .9995 .9997

.9988 .9991 .9994 .9996 .9997

.9988 .9992 .9994 .9996 .9997

.9989 .9992 .9994 .9996 .9997

.9989 .9992 .9994 .9996 .9997

.9989 .9992 .9995 .9996 .9997

.9990 .9993 .9995 .9996 .9997

.9990 .9993 .9995 .9997 .9998

NOTE: x  (a  m)/s where a is the observed value, m is the mean, and s is the standard deviation. x is the value in the left-hand column  the value in the top row. F(x) is the probability that a point will be less than or equal to x. F(x) is the value in the body of the table. Example: The probability that an observation will be less than or equal to 1.04 is .8508. NOTE: F(x)  1  F(x).

MATHEMATICAL TABLES

1-11

Table 1.1.13 Cumulative Chi-Square Distribution t x sn22d/2e2x/2 dx Fstd 5 3 n/2 0 2 [sn 2 2d/2]! F n 1 2 3 4 5

.005

.010

.025

.050

.100

.250

.500

.750

.900

.950

.975

.990

.995

.000039 .0100 .0717 .207 .412

.00016 .0201 .155 .297 .554

.00098 .0506 .216 .484 .831

.0039 .103 .352 .711 1.15

.0158 .211 .584 1.06 1.61

.101 .575 1.21 1.92 2.67

.455 1.39 2.37 3.36 4.35

1.32 2.77 4.11 5.39 6.63

2.70 4.61 6.25 7.78 9.24

3.84 5.99 7.81 9.49 11.1

5.02 7.38 9.35 11.1 12.8

6.62 9.21 11.3 13.3 15.1

7.86 10.6 12.8 14.9 16.7

5.35 6.35 7.34 8.34 9.34

7.84 9.04 10.2 11.4 12.5

10.6 12.0 13.4 14.7 16.0

12.6 14.1 15.5 16.9 18.3

14.4 16.0 17.5 19.0 20.5

16.8 18.5 20.1 21.7 23.2

18.5 20.3 22.0 23.6 25.2

6 7 8 9 10

.676 .989 1.34 1.73 2.16

.872 1.24 1.65 2.09 2.56

1.24 1.69 2.18 2.70 3.25

1.64 2.17 2.73 3.33 3.94

2.20 2.83 3.49 4.17 4.87

3.45 4.25 5.07 5.90 6.74

11 12 13 14 15

2.60 3.07 3.57 4.07 4.60

3.05 3.57 4.11 4.66 5.23

3.82 4.40 5.01 5.63 6.26

4.57 5.23 5.89 6.57 7.26

5.58 6.30 7.04 7.79 8.55

7.58 8.44 9.30 10.2 11.0

10.3 11.3 12.3 13.3 14.3

13.7 14.8 16.0 17.1 18.2

17.3 18.5 19.8 21.1 22.3

19.7 21.0 22.4 23.7 25.0

21.9 23.3 24.7 26.1 27.5

24.7 26.2 27.7 29.1 30.6

26.8 28.3 29.8 31.3 32.8

16 17 18 19 20

5.14 5.70 6.26 6.84 7.43

5.81 6.41 7.01 7.63 8.26

6.91 7.56 8.23 8.91 9.59

7.96 8.67 9.39 10.1 10.9

9.31 10.1 10.9 11.7 12.4

11.9 12.8 13.7 14.6 15.5

15.3 16.3 17.3 18.3 19.3

19.4 20.5 21.6 22.7 23.8

23.5 24.8 26.0 27.2 28.4

26.3 27.6 28.9 30.1 31.4

28.8 30.2 31.5 32.9 34.2

32.0 33.4 34.8 36.2 37.6

34.3 35.7 37.2 38.6 40.0

21 22 23 24 25

8.03 8.64 9.26 9.89 10.5

8.90 9.54 10.2 10.9 11.5

10.3 11.0 11.7 12.4 13.1

11.6 12.3 13.1 13.8 14.6

13.2 14.0 14.8 15.7 16.5

16.3 17.2 18.1 19.0 19.9

20.3 21.3 22.3 23.3 24.3

24.9 26.0 27.1 28.2 29.3

29.6 30.8 32.0 33.2 34.4

32.7 33.9 35.2 36.4 37.7

35.5 36.8 38.1 39.4 40.6

38.9 40.3 41.6 43.0 44.3

41.4 42.8 44.2 45.6 46.9

26 27 28 29 30

11.2 11.8 12.5 13.1 13.8

12.2 12.9 13.6 14.3 15.0

13.8 14.6 15.3 16.0 16.8

15.4 16.2 16.9 17.7 18.5

17.3 18.1 18.9 19.8 20.6

20.8 21.7 22.7 23.6 24.5

25.3 26.3 27.3 28.3 29.3

30.4 31.5 32.6 33.7 34.8

35.6 36.7 37.9 39.1 40.3

38.9 40.1 41.3 42.6 43.8

41.9 43.2 44.5 45.7 47.0

45.6 47.0 48.3 49.6 50.9

48.3 49.6 51.0 52.3 53.7

NOTE: n is the number of degrees of freedom. Values for t are in the body of the table. Example: The probability that, with 16 degrees of freedom, a point will be 23.5 is .900.

1-12

MATHEMATICAL TABLES Table 1.1.14 t

Fstd 5 3

2`

Cumulative “Student’s” Distribution n21 a b! 2 dx n22 x 2 sn11d/2 a b ! 2pn a1 1 b 2 n

F n

.75

.90

.95

.975

.99

.995

.9995

1 2 3 4 5

1.000 .816 .765 .741 .727

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.70 4.303 3.182 2.776 2.571

31.82 6.965 4.541 3.747 3.365

63.66 9.925 5.841 4.604 4.032

636.3 31.60 12.92 8.610 6.859

6 7 8 9 10

.718 .711 .706 .703 .700

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

5.959 5.408 5.041 4.781 4.587

11 12 13 14 15

.697 .695 .694 .692 .691

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

4.437 4.318 4.221 4.140 4.073

16 17 18 19 20

.690 .689 .688 .688 .687

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

4.015 3.965 3.922 3.883 3.850

21 22 23 24 25

.686 .686 .685 .685 .684

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

3.819 3.792 3.768 3.745 3.725

26 27 28 29 30

.684 .684 .683 .683 .683

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

3.707 3.690 3.674 3.659 3.646

40 60 120

.681 .679 .677

1.303 1.296 1.289

1.684 1.671 1.658

2.021 2.000 1.980

2.423 2.390 2.385

2.704 2.660 2.617

3.551 3.460 3.373

NOTE: n is the number of degrees of freedom. Values for t are in the body of the table. Example: The probability that, with 16 degrees of freedom, a point will be 2.921 is .995. NOTE: F(t)  1  F(t).

MATHEMATICAL TABLES

1-13

Table 1.1.15 Cumulative F Distribution m degrees of freedom in numerator; n in denominator F [sm 1 n 2 2d/2]!m m/2nn/2x sm22d/2 sn 1 mxd2sm1nd/2 dx GsFd 5 3 [sm 2 2d/2]![sn 2 2d/2]! 0 Upper 5% points (F.95)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120



1 2 3 4 5

161 18.5 10.1 7.71 6.61

200 19.0 9.55 6.94 5.79

216 19.2 9.28 6.59 5.41

225 19.2 9.12 6.39 5.19

230 19.3 9.01 6.26 5.05

234 19.3 8.94 6.16 4.95

237 19.4 8.89 6.09 4.88

239 19.4 8.85 6.04 4.82

241 19.4 8.81 6.00 4.77

242 19.4 8.79 5.96 4.74

244 19.4 8.74 5.91 4.68

246 19.4 8.70 5.86 4.62

248 19.4 8.66 5.80 4.56

249 19.5 8.64 5.77 4.53

250 19.5 8.62 5.75 4.50

251 19.5 8.59 5.72 4.46

252 19.5 8.57 5.69 4.43

253 19.5 8.55 5.66 4.40

254 19.5 8.53 5.63 4.37

6 7 8 9 10

5.99 5.59 5.32 5.12 4.96

5.14 4.74 4.46 4.26 4.10

4.76 4.35 4.07 3.86 3.71

4.53 4.12 3.84 3.63 3.48

4.39 3.97 3.69 3.48 3.33

4.28 3.87 3.58 3.37 3.22

4.21 3.79 3.50 3.29 3.14

4.15 3.73 3.44 3.23 3.07

4.10 3.68 3.39 3.18 3.02

4.06 3.64 3.35 3.14 2.98

4.00 3.57 3.28 3.07 2.91

3.94 3.51 3.22 3.01 2.85

3.87 3.44 3.15 2.94 2.77

3.84 3.41 3.12 2.90 2.74

3.81 3.38 3.08 2.86 2.70

3.77 3.34 3.04 2.83 2.66

3.74 3.30 3.01 2.79 2.62

3.70 3.27 2.97 2.75 2.58

3.67 3.23 2.93 2.71 2.54

11 12 13 14 15

4.84 4.75 4.67 4.60 4.54

3.98 3.89 3.81 3.74 3.68

3.59 3.49 3.41 3.34 3.29

3.36 3.26 3.18 3.11 3.06

3.20 3.11 3.03 2.96 2.90

3.09 3.00 2.92 2.85 2.79

3.01 2.91 2.83 2.76 2.71

2.95 2.85 2.77 2.70 2.64

2.90 2.80 2.71 2.65 2.59

2.85 2.75 2.67 2.60 2.54

2.79 2.69 2.60 2.53 2.48

2.72 2.62 2.53 2.46 2.40

2.65 2.54 2.46 2.39 2.33

2.61 2.51 2.42 2.35 2.29

2.57 2.47 2.38 2.31 2.25

2.53 2.43 2.34 2.27 2.20

2.49 2.38 2.30 2.22 2.16

2.45 2.34 2.25 2.18 2.11

2.40 2.30 2.21 2.13 2.07

16 17 18 19 20

4.49 4.45 4.41 4.38 4.35

3.63 3.59 3.55 3.52 3.49

3.24 3.20 3.16 3.13 3.10

3.01 2.96 2.93 2.90 2.87

2.85 2.81 2.77 2.74 2.71

2.74 2.70 2.66 2.63 2.60

2.66 2.61 2.58 2.54 2.51

2.59 2.55 2.51 2.48 2.45

2.54 2.49 2.46 2.42 2.39

2.49 2.45 2.41 2.38 2.35

2.42 2.38 2.34 2.31 2.28

2.35 2.31 2.27 2.23 2.20

2.28 2.23 2.19 2.16 2.12

2.24 2.19 2.15 2.11 2.08

2.19 2.15 2.11 2.07 2.04

2.15 2.10 2.06 2.03 1.99

2.11 2.06 2.02 1.98 1.95

2.06 2.01 1.97 1.93 1.90

2.01 1.96 1.92 1.88 1.84

21 22 23 24 25

4.32 4.30 4.28 4.26 4.24

3.47 3.44 3.42 3.40 3.39

3.07 3.05 3.03 3.01 2.99

2.84 2.82 2.80 2.78 2.76

2.68 2.66 2.64 2.62 2.60

2.57 2.55 2.53 2.51 2.49

2.49 2.46 2.44 2.42 2.40

2.42 2.40 2.37 2.36 2.34

2.37 2.34 2.32 2.30 2.28

2.32 2.30 2.27 2.25 2.24

2.25 2.23 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.10 2.07 2.05 2.03 2.01

2.05 2.03 2.01 1.98 1.96

2.01 1.98 1.96 1.94 1.92

1.96 1.94 1.91 1.89 1.87

1.92 1.89 1.86 1.84 1.82

1.87 1.84 1.81 1.79 1.77

1.81 1.78 1.76 1.73 1.71

30 40 60 120 

4.17 4.08 4.00 3.92 3.84

3.32 3.23 3.15 3.07 3.00

2.92 2.84 2.76 2.68 2.60

2.69 2.61 2.53 2.45 2.37

2.53 2.45 2.37 2.29 2.21

2.42 2.34 2.25 2.18 2.10

2.33 2.25 2.17 2.09 2.01

2.27 2.18 2.10 2.02 1.94

2.21 2.12 2.04 1.96 1.88

2.16 2.08 1.99 1.91 1.83

2.09 2.00 1.92 1.83 1.75

2.01 1.92 1.84 1.75 1.67

1.93 1.84 1.75 1.66 1.57

1.89 1.79 1.70 1.61 1.52

1.84 1.74 1.65 1.55 1.46

1.79 1.69 1.59 1.50 1.39

1.74 1.64 1.53 1.43 1.32

1.68 1.58 1.47 1.35 1.22

1.62 1.51 1.39 1.25 1.00

Upper 1% points (F.99)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120



1 2 3 4 5

4052 98.5 34.1 21.2 16.3

5000 99.0 30.8 18.0 13.3

5403 99.2 29.5 16.7 12.1

5625 99.2 28.7 16.0 11.4

5764 99.3 28.2 15.5 11.0

5859 99.3 27.9 15.2 10.7

5928 99.4 27.7 15.0 10.5

5982 99.4 27.5 14.8 10.3

6023 99.4 27.3 14.7 10.2

6056 99.4 27.2 14.5 10.1

6106 99.4 27.1 14.4 9.89

6157 99.4 26.9 14.2 9.72

6209 99.4 26.7 14.0 9.55

6235 99.5 26.6 13.9 9.47

6261 99.5 26.5 13.8 9.38

6287 99.5 26.4 13.7 9.29

6313 99.5 26.3 13.7 9.20

6339 99.5 26.2 13.6 9.11

6366 99.5 26.1 13.5 9.02

6 7 8 9 10

13.7 12.2 11.3 10.6 10.0

!0.9 9.55 8.65 8.02 7.56

9.78 8.45 7.59 6.99 6.55

9.15 7.85 7.01 6.42 5.99

8.75 7.46 6.63 6.06 5.64

8.47 7.19 6.37 5.80 5.39

8.26 6.99 6.18 5.61 5.20

8.10 6.84 6.03 5.47 5.06

7.98 6.72 5.91 5.35 4.94

7.87 6.62 5.81 5.26 4.85

7.72 6.47 5.67 5.11 4.71

7.56 6.31 5.52 4.96 4.56

7.40 6.16 5.36 4.81 4.41

7.31 6.07 5.28 4.73 4.33

7.23 5.99 5.20 4.65 4.25

7.14 5.91 5.12 4.57 4.17

7.06 5.82 5.03 4.48 4.08

6.97 5.74 4.95 4.40 4.40

6.88 5.65 4.86 4.31 3.91

11 12 13 14 15

9.65 9.33 9.07 8.86 8.68

7.21 6.93 6.70 6.51 6.36

6.22 5.95 5.74 5.56 5.42

5.67 5.41 5.21 5.04 4.89

5.32 5.06 4.86 4.70 4.56

5.07 4.82 4.62 4.46 4.32

4.89 4.64 4.44 4.28 4.14

4.74 4.50 4.30 4.14 4.00

4.63 4.39 4.19 4.03 3.89

4.54 4.30 4.10 3.94 3.80

4.40 4.16 3.96 3.80 3.67

4.25 4.01 3.82 3.66 3.52

4.10 3.86 3.66 3.51 3.37

4.02 3.78 3.59 3.43 3.29

3.94 3.70 3.51 3.35 3.21

3.86 3.62 3.43 3.27 3.13

3.78 3.54 3.34 3.18 3.05

3.69 3.45 3.25 3.09 2.96

3.60 3.36 3.17 3.00 2.87

16 17 18 19 20

8.53 8.40 8.29 8.19 8.10

6.23 6.11 6.01 5.93 5.85

5.29 5.19 5.09 5.01 4.94

4.77 4.67 4.58 4.50 4.43

4.44 4.34 4.25 4.17 4.10

4.20 4.10 4.01 3.94 3.87

4.03 3.93 3.84 3.77 3.70

3.89 3.79 3.71 3.63 3.56

3.78 3.68 3.60 3.52 3.46

3.69 3.59 3.51 3.43 3.37

3.55 3.46 3.37 3.30 3.23

3.41 3.31 3.23 3.15 3.09

3.26 3.16 3.08 3.00 2.94

3.18 3.08 3.00 2.92 2.86

3.10 3.00 2.92 2.84 2.78

3.02 2.92 2.84 2.76 2.69

2.93 2.83 2.75 2.67 2.61

2.84 2.75 2.66 2.58 2.52

2.75 2.65 2.57 2.49 2.42

21 22 23 24 25

8.02 7.95 7.88 7.82 7.77

5.78 5.72 5.66 5.61 5.57

4.87 4.82 4.76 4.72 4.68

4.37 4.31 4.26 4.22 4.18

4.04 3.99 3.94 3.90 3.86

3.81 3.76 3.71 3.67 3.63

3.64 3.59 3.54 3.50 3.46

3.51 3.45 3.41 3.36 3.32

3.40 3.35 3.30 3.26 3.22

3.31 3.26 3.21 3.17 3.13

3.17 3.12 3.07 3.03 2.99

3.03 2.98 2.93 2.89 2.85

2.88 2.83 2.78 2.74 2.70

2.80 2.75 2.70 2.66 2.62

2.72 2.67 2.62 2.58 2.53

2.64 2.58 2.54 2.49 2.45

2.55 2.50 2.45 2.40 2.36

2.46 2.40 2.35 2.31 2.27

2.36 2.31 2.26 2.21 2.17

30 40 60 120 

7.56 7.31 7.08 6.85 6.63

5.39 5.18 4.98 4.79 4.61

4.51 4.31 4.13 3.95 3.78

4.02 3.83 3.65 3.48 3.32

3.70 3.51 3.34 3.17 3.02

3.47 3.29 3.12 2.96 2.80

3.30 3.12 2.95 2.79 2.64

3.17 2.99 2.82 2.66 2.51

3.07 2.89 2.72 2.56 2.41

2.98 2.80 2.63 2.47 2.32

2.84 2.66 2.50 2.34 2.18

2.70 2.52 2.35 2.19 2.04

2.55 2.37 2.20 2.03 1.88

2.47 2.29 2.12 1.95 1.79

2.39 2.20 2.03 1.86 1.70

2.30 2.11 1.94 1.76 1.59

2.21 2.02 1.84 1.66 1.47

2.11 1.92 1.73 1.53 1.32

2.01 1.80 1.60 1.38 1.00

NOTE: m is the number of degrees of freedom in the numerator of F; n is the number of degrees of freedom in the denominator of F. Values for F are in the body of the table. G is the probability that a point, with m and n degrees of freedom will be F. Example: With 2 and 5 degrees of freedom, the probability that a point will be 13.3 is .99. SOURCE: “Chemical Engineers’ Handbook,” 5th edition, by R. H. Perry and C. H. Chilton, McGraw-Hill, 1973. Used with permission.

1-14

MATHEMATICAL TABLES Table 1.1.16

Standard Distribution of Residuals

a  any positive quantity y  the number of residuals which are numerically a r  the probable error of a single observation n  number of observations

Table 1.1.17

a r

y n

0.0 1 2 3 4

.000 .054 .107 .160 .213

0.5 6 7 8 9

.264 .314 .363 .411 .456

1.0 1 2 3 4

.500 .542 .582 .619 .655

1.5 6 7 8 9

.688 .719 .748 .775 .800

2.0 1 2 3 4

.823 .843 .862 .879 .895

Diff 54 53 53 53 51 50 49 48 45 44 42 40 37 36 33 31 29 27 25 23

a r

y n

2.5 6 7 8 9

.908 .921 .931 .941 .950

3.0 1 2 3 4

.957 .963 .969 .974 .978

3.5 6 7 8 9

.982 .985 .987 .990 .991

4.0

.993

5.0

.999

Diff 13 10 10 9 7 6 6 5 4 4 3 2 3 1 2 6

20 19 17 16 13

Factors for Computing Probable Error Bessel

n

Peters

Bessel

n

0.6745 !sn 2 1d

0.6745 !nsn 2 1d

0.8453 !nsn 2 1d

0.8453 n!n 2 1

2 3 4

.6745 .4769 .3894

.4769 .2754 .1947

.5978 .3451 .2440

.4227 .1993 .1220

5 6 7 8 9

.3372 .3016 .2754 .2549 .2385

.1508 .1231 .1041 .0901 .0795

.1890 .1543 .1304 .1130 .0996

.0845 .0630 .0493 .0399 .0332

10 11 12 13 14

.2248 .2133 .2034 .1947 .1871

.0711 .0643 .0587 .0540 .0500

.0891 .0806 .0736 .0677 .0627

.0282 .0243 .0212 .0188 .0167

15 16 17 18 19

.1803 .1742 .1686 .1636 .1590

.0465 .0435 .0409 .0386 .0365

.0583 .0546 .0513 .0483 .0457

.0151 .0136 .0124 .0114 .0105

20 21 22 23 24

.1547 .1508 .1472 .1438 .1406

.0346 .0329 .0314 .0300 .0287

.0434 .0412 .0393 .0376 .0360

.0097 .0090 .0084 .0078 .0073

25 26 27 28 29

.1377 .1349 .1323 .1298 .1275

.0275 .0265 .0255 .0245 .0237

.0345 .0332 .0319 .0307 .0297

.0069 .0065 .0061 .0058 .0055

Peters

0.6745 !sn 2 1d

0.6745 !nsn 2 1d

0.8453 !nsn 2 1d

0.8453 n!n 2 1

30 31 32 33 34

.1252 .1231 .1211 .1192 .1174

.0229 .0221 .0214 .0208 .0201

.0287 .0277 .0268 .0260 .0252

.0052 .0050 .0047 .0045 .0043

35 36 37 38 39

.1157 .1140 .1124 .1109 .1094

.0196 .0190 .0185 .0180 .0175

.0245 .0238 .0232 .0225 .0220

.0041 .0040 .0038 .0037 .0035

40 45

.1080 .1017

.0171 .0152

.0214 .0190

.0034 .0028

50 55

.0964 .0918

.0136 .0124

.0171 .0155

.0024 .0021

60 65

.0878 .0843

.0113 .0105

.0142 .0131

.0018 .0016

70 75

.0812 .0784

.0097 .0091

.0122 .0113

.0015 .0013

80 85

.0759 .0736

.0085 .0080

.0106 .0100

.0012 .0011

90 95

.0715 .0696

.0075 .0071

.0094 .0089

.0010 .0009

100

.0678

.0068

.0085

.0008

MATHEMATICAL TABLES Table 1.1.18

1-15

Decimal Equivalents Common fractions

From minutes and seconds into decimal parts of a degree 0r 1 2 3 4 5r 6 7 8 9 10r 1 2 3 4 15r 6 7 8 9 20r 1 2 3 4 25r 6 7 8 9 30r 1 2 3 4 35r 6 7 8 9 40r 1 2 3 4 45r 6 7 8 9 50r 1 2 3 4 55r 6 7 8 9 60r

08.0000 .0167 .0333 .05 .0667 .0833 .10 .1167 .1333 .15 08.1667 .1833 .20 .2167 .2333 .25 .2667 .2833 .30 .3167 08.3333 .35 .3667 .3833 .40 .4167 .4333 .45 .4667 .4833 08.50 .5167 .5333 .55 .5667 .5833 .60 .6167 .6333 .65 08.6667 .6833 .70 .7167 .7333 .75 .7667 .7833 .80 .8167 08.8333 .85 .8667 .8833 .90 .9167 .9333 .95 .9667 .9833 1.00

0s 1 2 3 4 5s 6 7 8 9 10s 1 2 3 4 15s 6 7 8 9 20s 1 2 3 4 25s 6 7 8 9 30s 1 2 3 4 35s 6 7 8 9 40s 1 2 3 4 45s 6 7 8 9 50s 1 2 3 4 55s 6 7 8 9 60s

From decimal parts of a degree into minutes and seconds (exact values) 08.0000 .0003 .0006 .0008 .0011 .0014 .0017 .0019 .0022 .0025 08.0028 .0031 .0033 .0036 .0039 .0042 .0044 .0047 .005 .0053 08.0056 .0058 .0061 .0064 .0067 .0069 .0072 .0075 .0078 .0081 08.0083 .0086 .0089 .0092 .0094 .0097 .01 .0103 .0106 .0108 08.0111 .0114 .0117 .0119 .0122 .0125 .0128 .0131 .0133 .0136 08.0139 .0142 .0144 .0147 .015 .0153 .0156 .0158 .0161 .0164 08.0167

08.00 1 2 3 4 08.05 6 7 8 9 08.10 1 2 3 4 08.15 6 7 8 9 08.20 1 2 3 4 08.25 6 7 8 9 08 .30 1 2 3 4 08.35 6 7 8 9 08.40 1 2 3 4 08.45 6 7 8 9 08.50

0r 0r 36s 1r 12s 1r 48s 2r 24s 3r 3r 36s 4r 12s 4r 48s 5r 24s 6r 6r 36s 7r 12s 7r 48s 8r 24s 9r 9r 36s 10r 12s 10r 48s 11r 24s 12r 12r 36s 13r 12s 13r 48s 14r 24s 15r 15r 36s 16r 12s 16r 48s 17r 24s 18r 18r 36s 19r 12s 19r 48s 20r 24s 21r 21r 36s 22r 12s 22r 48s 23r 24s 24r 24r 36s 25r 12s 25r 48s 26r 24s 27r 27r 36s 28r 12s 28r 48s 29r 24s 30r

08.50 1 2 3 4 08.55 6 7 8 9 08.60 1 2 3 4 08.65 6 7 8 9 08.70 1 2 3 4 08.75 6 7 8 9 08.80 1 2 3 4 08.85 6 7 8 9 08.90 1 2 3 4 08.95 6 7 8 9 18.00

08.000 1 2 3 4 08.005 6 7 8 9 08.010

0s.0 3s.6 7s.2 10s.8 14s.4 18s 21s.6 25s.2 28s.8 32s.4 36s

8 ths 30r 30r 36s 31r 12s 31r 48s 32r 24s 33r 33r 36s 34r 12s 34r 48s 35r 24s 36r 36r 36s 37r 12s 37r 48s 38r 24s 39r 39r 36s 40r 12s 40r 48s 41r 24s 42r 42r 36s 43r 12s 43r 48s 44r 24s 45r 45r 36s 46r 12s 46r 48s 47r 24s 48r 48r 36s 49r 12s 49r 48s 50r 24s 51r 51r 36s 52r 12s 52r 48s 53r 24s 54r 54r 36s 55r 12s 55r 48s 56r 24s 57r 57r 36s 58r 12s 58r 48s 59r 24s 60r

16 ths

32 nds 1

1

2 3

1

2

4 5

3

6 7

2

4

8 9

5

10 11

3

6

12 13

7

14 15

4

8

16 17

9

18 19

5

10

20 21

11

22 23

6

12

24 25

13

26 27

7

14

28 29

15

30 31

64 ths

Exact decimal values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

.01 5625 .03 125 .04 6875 .06 25 .07 8125 .09 375 .10 9375 .12 5 .14 0625 .15 625 .17 1875 .18 75 .20 3125 .21 875 .23 4375 .25 .26 5625 .28 125 .29 6875 .31 25 .32 8125 .34 375 .35 9375 .37 5 .39 0625 .40 625 .42 1875 .43 75 .45 3125 .46 875 .48 4375 .50 .51 5625 .53 125 .54 6875 .56 25 .57 8125 .59 375 .60 9375 .62 5 .64 0625 .65 625 .67 1875 .68 75 .70 3125 .71 875 .73 4375 .75 .76 5625 .78 125 .79 6875 .81 25 .82 8125 .84 375 .85 9375 .87 5 .89 0625 .90 625 .92 1875 .93 75 .95 3125 .96 875 .98 4375

1.2 MEASURING UNITS by John T. Baumeister REFERENCES: “International Critical Tables,” McGraw-Hill. “Smithsonian Physical Tables,” Smithsonian Institution. “Landolt-Börnstein: Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik,” Springer. “Handbook of Chemistry and Physics,” Chemical Rubber Co. “Units and Systems of Weights and Measures; Their Origin, Development, and Present Status,” NBS LC 1035 (1976). “Weights and Measures Standards of the United States, a Brief History,” NBS Spec. Pub. 447 (1976). “Standard Time,” Code of Federal Regulations, Title 49. “Fluid Meters, Their Theory and Application,” 6th ed., chaps. 1–2, ASME, 1971. H.E. Huntley, “Dimensional Analysis,” Richard & Co., New York, 1951. “U.S. Standard Atmosphere, 1962,” Government Printing Office. Public Law 89-387, “Uniform Time Act of 1966.” Public Law 94-168, “Metric Conversion Act of 1975.” ASTM E380-91a, “Use of the International Standards of Units (SI) (the Modernized Metric System).” The International System of Units,” NIST Spec. Pub. 330. “Guide for the Use of the International System of Units (SI),” NIST Spec. Pub. 811. “Guidelines for Use of the Modernized Metric System,” NBS LC 1120. “NBS Time and Frequency Dissemination Services,” NBS Spec. Pub. 432. “Factors for High Precision Conversion,” NBS LC 1071. American Society of Mechanical Engineers SI Series, ASME SI 19. Jespersen and FitzRandolph, “From Sundials to Atomic Clocks: Understanding Time and Frequency,” NBS, Monograph 155. ANSI/IEEE Std 268-1992, “American National Standard for Metric Practice.”

U.S. CUSTOMARY SYSTEM (USCS)

The USCS, often called the “inch-pound system,” is the system of units most commonly used for measures of weight and length (Table 1.2.1). The units are identical for practical purposes with the corresponding English units, but the capacity measures differ from those used in the British Commonwealth, the U.S. gallon being defined as 231 cu in and the bushel as 2,150.42 cu in, whereas the corresponding British Imperial units are, respectively, 277.42 cu in and 2,219.36 cu in (1 Imp gal  1.2 U.S. gal, approx; 1 Imp bu  1.03 U.S. bu, approx).

Table 1.2.1

U.S. Customary Units Units of length

12 inches 3 feet 51⁄2 yards  161⁄2 feet 40 poles  220 yards 8 furlongs  1,760 yards  5,280 feet 3 miles 4 inches 9 inches 6,076.11549 feet 6 feet 120 fathoms 1 nautical mile per hr

j

 1 foot  1 yard  1 rod, pole, or perch  1 furlong  1 mile  1 league  1 hand  1 span Nautical units  1 international nautical mile  1 fathom  1 cable length  1 knot

Surveyor’s or Gunter’s units 7.92 inches  1 link 100 links  66 ft  4 rods  1 chain 80 chains  1 mile 331⁄3 inches  1 vara (Texas) Units of area 144 square inches 9 square feet 301⁄4 square yards 1-16

 1 square foot  1 square yard  1 square rod, pole, or perch

160 square rods  10 square chains  43,560 square feet  5,645 sq varas (Texas)

J

640 acres  1 square mile  1 circular inch  area of circle 1 inch in diameter 1 square inch 1 circular mil 1,000,000 cir mils

j

 1 acre

h

1 “section” of U.S. government-surveyed land

 0.7854 sq in  1.2732 circular inches  area of circle 0.001 in in diam  1 circular inch Units of volume

1,728 cubic inches 231 cubic inches 27 cubic feet 1 cord of wood 1 perch of masonry

 1 cubic foot  1 gallon  1 cubic yard  128 cubic feet  161⁄2 to 25 cu ft

Liquid or fluid measurements 4 gills  1 pint 2 pints  1 quart 4 quarts  1 gallon 7.4805 gallons  1 cubic foot (There is no standard liquid barrel; by trade custom, 1 bbl of petroleum oil, unrefined  42 gal. The capacity of the common steel barrel used for refined petroleum products and other liquids is 55 gal.) Apothecaries’ liquid measurements  1 liquid dram or drachm  1 liquid ounce  1 pint

60 minims 8 drams 16 ounces

Water measurements The miner’s inch is a unit of water volume flow no longer used by the Bureau of Reclamation. It is used within particular water districts where its value is defined by statute. Specifically, within many of the states of the West the miner’s inch is 1⁄50 cubic foot per second. In others it is equal to 1⁄40 cubic foot per second, while in the state of Colorado, 38.4 miner’s inch is equal to 1 cubic-foot per second. In SI units, these correspond to .32  106 m3/s, .409  106 m3/s, and .427  106 m3/s, respectively. Dry measures 2 pints  1 quart 8 quarts  1 peck 4 pecks  1 bushel 1 std bbl for fruits and vegetables  7,056 cu in or 105 dry qt, struck measure 1 Register ton 1 U.S. shipping ton 1 British shipping ton

Shipping measures  100 cu ft  40 cu ft  32.14 U.S. bu or 31.14 Imp bu  42 cu ft  32.70 Imp bu or 33.75 U.S. bu

Board measurements (Based on nominal not actual dimensions; see Table 12.2.8) 144 cu in  volume of board 1 board foot  1 ft sq and 1 in thick

h

The international log rule, based upon 1⁄4 in kerf, is expressed by the formula X  0.904762(0.22 D2  0.71 D) where X is the number of board feet in a 4-ft section of a log and D is the top diam in in. In computing the number of board feet in a log, the taper is taken at 1⁄2 in per 4 ft linear, and separate computation is made for each 4-ft section.

THE INTERNATIONAL SYSTEM OF UNITS (SI) Weights (The grain is the same in all systems.) 16 drams  437.5 grains 16 ounces  7,000 grains 100 pounds 2,000 pounds 2,240 pounds 1 std lime bbl, small 1 std lime bbl, large Also (in Great Britain): 14 pounds 2 stone  28 pounds 4 quarters  112 pounds 20 hundredweight

Avoirdupois weights  1 ounce  1 pound  1 cental  1 short ton  1 long ton  180 lb net  280 lb net  1 stone  1 quarter  1 hundredweight (cwt)  1 long ton

Troy weights 24 grains  1 pennyweight (dwt) 20 pennyweights  480 grains  1 ounce 12 ounces  5,760 grains  1 pound 1 assay ton  29,167 milligrams, or as many milligrams as there are troy ounces in a ton of 2,000 lb avoirdupois. Consequently, the number of milligrams of precious metal yielded by an assay ton of ore gives directly the number of troy ounces that would be obtained from a ton of 2,000 lb avoirdupois. 20 grains 3 scruples  60 grains 8 drams 12 ounces  5,760 grains

Apothecaries’ weights  1 scruple  1 dram  1 ounce  1 pound

Weight for precious stones 1 carat  200 milligrams (Used by almost all important nations) 60 seconds 60 minutes 90 degrees 360 degrees 57.2957795 degrees ( 5717r44.806s)

Circular measures  1 minute  1 degree  1 quadrant  circumference  1 radian (or angle having arc of length equal to radius)

METRIC SYSTEM

In the United States the name “metric system” of length and mass units is commonly taken to refer to a system that was developed in France about 1800. The unit of length was equal to 1/10,000,000 of a quarter meridian (north pole to equator) and named the metre. A cube 1/10th metre on a side was the litre, the unit of volume. The mass of water filling this cube was the kilogram, or standard of mass; i.e., 1 litre of water  1 kilogram of mass. Metal bars and weights were constructed conforming to these prescriptions for the metre and kilogram. One bar and one weight were selected to be the primary representations. The kilogram and the metre are now defined independently, and the litre, although for many years defined as the volume of a kilogram of water at the temperature of its maximum density, 48C, and under a pressure of 76 cm of mercury, is now equal to 1 cubic decimeter. In 1866, the U.S. Congress formally recognized metric units as a legal system, thereby making their use permissible in the United States. In 1893, the Office of Weights and Measures (now the National Bureau of Standards), by executive order, fixed the values of the U.S. yard and pound in terms of the meter and kilogram, respectively, as 1 yard  3,600/3,937 m; and 1 lb  0.453 592 4277 kg. By agreement in 1959 among the national standards laboratories of the English-speaking nations, the relations in use now are: 1 yd  0.9144 m, whence 1 in 

1-17

25.4 mm exactly; and 1 lb  0.453 592 37 kg, or 1 lb  453.59 g (nearly).

THE INTERNATIONAL SYSTEM OF UNITS (SI)

In October 1960, the Eleventh General (International) Conference on Weights and Measures redefined some of the original metric units and expanded the system to include other physical and engineering units. This expanded system is called, in French, Le Système International d’Unités (abbreviated SI), and in English, The International System of Units.

The Metric Conversion Act of 1975 codifies the voluntary conversion of the U.S. to the SI system. It is expected that in time all units in the United States will be in SI form. For this reason, additional tables of units, prefixes, equivalents, and conversion factors are included below (Tables 1.2.2 and 1.2.3). SI consists of seven base units, two supplementary units, a series of derived units consistent with the base and supplementary units, and a series of approved prefixes for the formation of multiples and submultiples of the various units (see Tables 1.2.2 and 1.2.3). Multiple and submultiple prefixes in steps of 1,000 are recommended. (See ASTM E380-91a for further details.) Base and supplementary units are defined [NIST Spec. Pub. 330 (2001)] as: Metre The metre is defined as the length of path traveled by light in a vacuum during a time interval 1/299 792 458 of a second. Kilogram The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. Second The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2  107 newton per metre of length. Kelvin The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Mole The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. (When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.) Candela The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540  1012 hertz and that has a radiant intensity in that direction of 1⁄683 watt per steradian. Radian The unit of measure of a plane angle with its vertex at the center of a circle and subtended by an arc equal in length to the radius. Steradian The unit of measure of a solid angle with its vertex at the center of a sphere and enclosing an area of the spherical surface equal to that of a square with sides equal in length to the radius. SI conversion factors are listed in Table 1.2.4 alphabetically (adapted from ASTM E380-91a, “Standard Practice for Use of the International System of Units (SI) (the Modernized Metric System).” Conversion factors are written as a number greater than one and less than ten with six or fewer decimal places. This number is followed by the letter E (for exponent), a plus or minus symbol, and two digits which indicate the power of 10 by which the number must be multiplied to obtain the correct value. For example: 3.523 907 E  02 is 3.523 907  102 or 0.035 239 07 An asterisk (*) after the sixth decimal place indicates that the conversion factor is exact and that all subsequent digits are zero. All other conversion factors have been rounded off.

1-18

MEASURING UNITS Table 1.2.2

SI Units

Quantity

Unit

SI symbol

Formula

Base units* Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity

metre kilogram second ampere kelvin mole candela

Plane angle Solid angle

radian steradian

m kg s A K mol cd Supplementary units* rad sr Derived units*

Acceleration Activity (of a radioactive source) Angular acceleration Angular velocity Area Density Electric capacitance Electrical conductance Electric field strength Electric inductance Electric potential difference Electric resistance Electromotive force Energy Entropy Force Frequency Illuminance Luminance Luminous flux Magnetic field strength Magnetic flux Magnetic flux density Magnetic potential difference Power Pressure Quantity of electricity Quantity of heat Radiant intensity Specific heat capacity Stress Thermal conductivity Velocity Viscosity, dynamic Viscosity, kinematic Voltage Volume Wave number Work

metre per second squared disintegration per second radian per second squared radian per second square metre kilogram per cubic metre farad siemens volt per metre henry volt ohm volt joule joule per kelvin newton hertz lux candela per square metre lumen ampere per metre weber tesla ampere watt pascal coulomb joule watt per steradian joule per kilogram-kelvin pascal watt per metre-kelvin metre per second pascal-second square metre per second volt cubic metre reciprocal metre joule

F S H V V J N Hz lx lm Wb T A W Pa C J

m/s2 (disintegration)/s rad/s2 rad/s m2 kg/m3 A  s/V A/V V/m V  s/A W/A V/A W/A Nm J/K kg  m/s2 1/s lm/m2 cd/m2 cd  sr A/m Vs Wb/m2

J

J/s N/m2 As Nm W/sr J/(kg  K) N/m2 W/(m  K) m/s Pa  s m2/s W/A m3 1/m Nm

min h d 8 r s L t u eV

1 min  60 s 1 h  60 min  3,600 s 1 d  24 h  86,400 s 18  p/180 rad 1r  (1⁄60)8  (p/10,800) rad 1s  (1⁄60)r  (p/648,000) rad 1 L  1 dm3  103 m3 1 t  103 kg 1 u  1.660 54  1027 kg 1 eV  1.602 18  1019 J

Pa

V

Units in use with the SI† Time

Plane angle

Volume Mass Energy

minute hour day degree minute‡ second‡ litre metric ton unified atomic mass unit§ electronvolt§

* ASTM E380-91a. † These units are not part of SI, but their use is both so widespread and important that the International Committee for Weights and Measures in 1969 recognized their continued use with the SI (see NIST Spec. Pub. 330). ‡ Use discouraged, except for special fields such as cartography. § Values in SI units obtained experimentally. These units are to be used in specialized fields only.

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.3

SI Prefixes*

Multiplication factors 1 000 000 000 000 000 000 000 000  10 1 000 000 000 000 000 000 000  1021 1 000 000 000 000 000 000  1018 1 000 000 000 000 000  1015 1 000 000 000 000  1012 1 000 000 000  109 1 000 000  106 1 000  103 100  102 10  101 0.1  101 0.01  102 0.001  103 0.000 001  106 0.000 000 001  109 0.000 000 000 001  1012 0.000 000 000 000 001  1015 0.000 000 000 000 000 001  1018 0.000 000 000 000 000 000 001  1021 0.000 000 000 000 000 000 000 001  1024 24

Prefix

SI symbol

yotta zetta exa peta tera giga mega kilo hecto† deka† deci† centi† milli micro nano pico femto atto zepto yocto

Y Z E P T G M k h da d c m m n P f a z y

* ANSI/IEEE Std 268-1992. † To be avoided where practical.

Table 1.2.4

SI Conversion Factors

To convert from abampere abcoulomb abfarad abhenry abmho abohm abvolt acre-foot (U.S. survey)a acre (U.S. survey)a ampere, international U.S. (AINTUS)b ampere, U.S. legal 1948 (AUS48) ampere-hour angstrom are astronomical unit atmosphere (normal) atmosphere (technical  1 kgf/cm2) bar barn barrel (for crude petroleum, 42 gal) board foot British thermal unit (International Table)c British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (398F) British thermal unit (598F) British thermal unit (608F) Btu (thermochemical)/foot2-second Btu (thermochemical)/foot2-minute Btu (thermochemical)/foot2-hour Btu (thermochemical)/inch2-second Btu (thermochemical)  in/s  ft2  8F (k, thermal conductivity) Btu (International Table)  in/s  ft2  8F (k, thermal conductivity) Btu (thermochemical)  in/h  ft2  8F (k, thermal conductivity) Btu (International Table)  in/h  ft2  8F (k, thermal conductivity) Btu (International Table)/ft2 Btu (thermochemical)/ft2 Btu (International Table)/h  ft2  8F (C, thermal conductance) Btu (thermochemical)/h  ft2  8F (C, thermal conductance) Btu (International Table)/pound-mass

to

Multiply by

ampere (A) coulomb (C) farad (F) henry (H) siemens (S) ohm ( ) volt (V) metre3 (m3) metre2 (m2) ampere (A) ampere (A) coulomb (C) metre (m) metre2 (m2) metre (m) pascal (Pa) pascal (Pa) pascal (Pa) metre2 (m2) metre3 (m3) metre3 (m3) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre-kelvin (W/m  K)

1.000 000*E01 1.000 000*E01 1.000 000*E09 1.000 000*E09 1.000 000*E09 1.000 000*E09 1.000 000*E08 1.233 489 E03 4.046 873 E03 9.998 43 E01 1.000 008 E00 3.600 000*E03 1.000 000*E10 1.000 000*E02 1.495 98 E11 1.013 25 E05 9.806 650*E04 1.000 000*E05 1.000 000*E28 1.589 873 E01 2.359 737 E03 1.055 056 E03 1.055 87 E03 1.054 350 E03 1.059 67 E03 1.054 80 E03 1.054 68 E03 1.134 893 E04 1.891 489 E02 3.152 481 E00 1.634 246 E06 5.188 732 E02

watt/metre-kelvin (W/m  K)

5.192 204 E02

watt/metre-kelvin (W/m  K)

1.441 314 E01

watt/metre-kelvin (W/m  K)

1.442 279 E01

joule/metre2 (J/m2) joule/metre2 (J/m2) watt/metre2-kelvin (W/m2  K)

1.135 653 E04 1.134 893 E04 5.678 263 E00

watt/metre2-kelvin (W/m2  K)

5.674 466 E00

joule/kilogram (J/kg)

2.326 000*E03

1-19

1-20

MEASURING UNITS Table 1.2.4

SI Conversion Factors

(Continued )

To convert from

to

Multiply by

Btu (thermochemical)/pound-mass Btu (International Table)/lbm  8F (c, heat capacity) Btu (thermochemical)/lbm  8F (c, heat capacity) Btu (International Table)/s  ft2  8F Btu (thermochemical)/s  ft2  8F Btu (International Table)/hour Btu (thermochemical)/second Btu (thermochemical)/minute Btu (thermochemical)/hour bushel (U.S.) calorie (International Table) calorie (mean) calorie (thermochemical) calorie (158C) calorie (208C) calorie (kilogram, International Table) calorie (kilogram, mean) calorie (kilogram, thermochemical) calorie (thermochemical)/centimetre2minute cal (thermochemical)/cm2 cal (thermochemical)/cm2  s cal (thermochemical)/cm  s  8C cal (International Table)/g cal (International Table)/g  8C cal (thermochemical)/g cal (thermochemical)/g  8C calorie (thermochemical)/second calorie (thermochemical)/minute carat (metric) centimetre of mercury (08C) centimetre of water (48C) centipoise centistokes chain (engineer or ramden) chain (surveyor or gunter) circular mil cord coulomb, international U.S. (CINTUS)b coulomb, U.S. legal 1948 (CUS48) cup curie day (mean solar) day (sidereal) degree (angle) degree Celsius degree centigrade degree Fahrenheit degree Fahrenheit deg F  h  ft2/Btu (thermochemical) (R, thermal resistance) deg F  h  ft2/Btu (International Table) (R, thermal resistance) degree Rankine dram (avoirdupois) dram (troy or apothecary) dram (U.S. fluid) dyne dyne-centimetre dyne-centimetre2 electron volt EMU of capacitance EMU of current EMU of electric potential EMU of inductance EMU of resistance ESU of capacitance ESU of current ESU of electric potential ESU of inductance

joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg  K)

2.324 444 E03 4.186 800*E03

joule/kilogram-kelvin (J/kg  K)

4.184 000*E03

watt/metre -kelvin (W/m  K) watt/metre2-kelvin (W/m2  K) watt (W) watt (W) watt (W) watt (W) metre3 (m3) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) watt/metre2 (W/m2)

2.044 175 E04 2.042 808 E04 2.930 711 E01 1.054 350 E03 1.757 250 E01 2.928 751 E01 3.523 907 E02 4.186 800*E00 4.190 02 E00 4.184 000*E00 4.185 80 E00 4.181 90 E00 4.186 800*E03 4.190 02 E03 4.184 000*E03 6.973 333 E02

joule/metre2 (J/m2) watt/metre2 (W/m2) watt/metre-kelvin (W/m  K) joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg  K) joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg  K) watt (W) watt (W) kilogram (kg) pascal (Pa) pascal (Pa) pascal-second (Pa  s) metre2/second (m2/s) meter (m) meter (m) metre2 (m2) metre3 (m3) coulomb (C)

4.184 000*E04 4.184 000*E04 4.184 000*E02 4.186 800*E03 4.186 800*E03 4.184 000*E03 4.184 000*E03 4.184 000*E00 6.973 333 E02 2.000 000*E04 1.333 22 E03 9.806 38 E01 1.000 000*E03 1.000 000*E06 3.048* E01 2.011 684 E01 5.067 075 E10 3.624 556 E00 9.998 43 E01

coulomb (C) metre3 (m3) becquerel (Bq) second (s) second (s) radian (rad) kelvin (K) kelvin (K) degree Celsius kelvin (K) kelvin-metre2/watt (K  m2/W)

1.000 008 E00 2.365 882 E04 3.700 000*E10 8.640 000 E04 8.616 409 E04 1.745 329 E02 tK  t8C  273.15 tK  t8C  273.15 t8C  (t8F  32)/1.8 tK  (t8F  459.67)/1.8 1.762 280 E01

kelvin-metre2/watt (K  m2/ W)

1.761 102 E01

kelvin (K) kilogram (kg) kilogram (kg) kilogram (kg) newton (N) newton-metre (N  m) pascal (Pa) joule (J) farad (F) ampere (A) volt (V) henry (H) ohm ( ) farad (F) ampere (A) volt (V) henry (H)

tK  t8R/1.8 1.771 845 E03 3.887 934 E03 3.696 691 E06 1.000 000*E05 1.000 000*E07 1.000 000*E01 1.602 18 E19 1.000 000*E09 1.000 000*E01 1.000 000*E08 1.000 000*E09 1.000 000*E09 1.112 650 E12 3.335 6 E10 2.997 9 E02 8.987 552 E11

2

2

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.4

SI Conversion Factors

To convert from ESU of resistance erg erg/centimetre2-second erg/second farad, international U.S. (FINTUS) faraday (based on carbon 12) faraday (chemical) faraday (physical) fathom (U.S. survey)a fermi (femtometer) fluid ounce (U.S.) foot foot (U.S. survey)a foot3/minute foot3/second foot3 (volume and section modulus) foot2 foot4 (moment of section)d foot/hour foot/minute foot/second foot2/second foot of water (39.28F) footcandle footcandle footlambert foot-pound-force foot-pound-force/hour foot-pound-force/minute foot-pound-force/second foot-poundal ft2/h (thermal diffusivity) foot/second2 free fall, standard furlong gal gallon (Canadian liquid) gallon (U.K. liquid) gallon (U.S. dry) gallon (U.S. liquid) gallon (U.S. liquid)/day gallon (U.S. liquid)/minute gamma gauss gilbert gill (U.K.) gill (U.S.) grade grade grain (1/7,000 lbm avoirdupois) gram gram/centimetre3 gram-force/centimetre2 hectare henry, international U.S. (HINTUS) hogshead (U.S.) horsepower (550 ft  lbf/s) horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (water) horsepower (U.K.) hour (mean solar) hour (sidereal) hundredweight (long) hundredweight (short) inch inch2 inch3 (volume and section modulus) inch3/minute inch4 (moment of section)d inch/second inch of mercury (328F)

(Continued ) to ohm ( ) joule (J) watt/metre2 (W/m2) watt (W) farad (F) coulomb (C) coulomb (C) coulomb (C) metre (m) metre (m) metre3 (m3) metre (m) metre (m) metre3/second (m3/s) metre3/second (m3/s) metre3 (m3) metre2 (m2) metre4 (m4) metre/second (m/s) metre/second (m/s) metre/second (m/s) metre2/second (m2/s) pascal (Pa) lumen/metre2 (lm/m2) lux (lx) candela/metre2 (cd/m2) joule (J) watt (W) watt (W) watt (W) joule (J) metre2/second (m2/s) metre/second2 (m/s2) metre/second2 (m/s2) metre (m) metre/second2 (m/s2) metre3 (m3) metre3 (m3) metre3 (m3) metre3 (m3) metre3/second (m3/s) metre3/second (m3/s) tesla (T) tesla (T) ampere-turn metre3 (m3) metre3 (m3) degree (angular) radian (rad) kilogram (kg) kilogram (kg) kilogram/metre3 (kg/m3) pascal (Pa) metre2 (m2) henry (H) metre3 (m3) watt (W) watt (W) watt (W) watt (W) watt (W) watt (W) second (s) second (s) kilogram (kg) kilogram (kg) metre (m) metre2 (m2) metre3 (m3) metre3/second (m3/s) metre4 (m4) metre/second (m/s) pascal (Pa)

Multiply by 8.987 552 E11 1.000 000*E07 1.000 000*E03 1.000 000*E07 9.995 05 E01 9.648 531 E04 9.649 57 E04 9.652 19 E04 1.828 804 E00 1.000 000*E15 2.957 353 E05 3.048 000*E01 3.048 006 E01 4.719 474 E04 2.831 685 E02 2.831 685 E02 9.290 304*E02 8.630 975 E03 8.466 667 E05 5.080 000*E03 3.048 000*E01 9.290 304*E02 2.988 98 E03 1.076 391 E01 1.076 391 E01 3.426 259 E00 1.355 818 E00 3.766 161 E04 2.259 697 E02 1.355 818 E00 4.214 011 E02 2.580 640*E05 3.048 000*E01 9.806 650*E00 2.011 68 *E02 1.000 000*E02 4.546 090 E03 4.546 092 E03 4.404 884 E03 3.785 412 E03 4.381 264 E08 6.309 020 E05 1.000 000*E09 1.000 000*E04 7.957 747 E01 1.420 653 E04 1.182 941 E04 9.000 000*E01 1.570 796 E02 6.479 891*E05 1.000 000*E03 1.000 000*E03 9.806 650*E01 1.000 000*E04 1.000 495 E00 2.384 809 E01 7.456 999 E02 9.809 50 E03 7.460 000*E02 7.354 99 E02 7.460 43 E02 7.457 0 E02 3.600 000*E03 3.590 170 E03 5.080 235 E01 4.535 924 E01 2.540 000*E02 6.451 600*E04 1.638 706 E05 2.731 177 E07 4.162 314 E07 2.540 000*E02 3.386 38 E03

1-21

1-22

MEASURING UNITS Table 1.2.4

SI Conversion Factors

(Continued )

To convert from inch of mercury (608F) inch of water (39.28F) inch of water (608F) inch/second2 joule, international U.S. (JINTUS)b joule, U.S. legal 1948 (JUS48) kayser kelvin kilocalorie (thermochemical)/minute kilocalorie (thermochemical)/second kilogram-force (kgf ) kilogram-force-metre kilogram-force-second2/metre (mass) kilogram-force/centimetre2 kilogram-force/metre3 kilogram-force/millimetre2 kilogram-mass kilometre/hour kilopond kilowatt hour kilowatt hour, international U.S. (kWhINTUS)b kilowatt hour, U.S. legal 1948 (kWhUS48) kip (1,000 lbf ) kip/inch2 (ksi) knot (international) lambert langley league, nautical (international and U.S.) league (U.S. survey)a league, nautical (U.K.) light year (365.2425 days) link (engineer or ramden) link (surveyor or gunter) litree lux maxwell mho microinch micron (micrometre) mil mile, nautical (international and U.S.) mile, nautical (U.K.) mile (international) mile (U.S. survey)a mile2 (international) mile2 (U.S. survey)a mile/hour (international) mile/hour (international) millimetre of mercury (08C) minute (angle) minute (mean solar) minute (sidereal) month (mean calendar) oersted ohm, international U.S. ( INT–US) ohm-centimetre ounce-force (avoirdupois) ounce-force-inch ounce-mass (avoirdupois) ounce-mass (troy or apothecary) ounce-mass/yard2 ounce (avoirdupois)(mass)/inch3 ounce (U.K. fluid) ounce (U.S. fluid) parsec peck (U.S.) pennyweight perm (08C) perm (23 8C)

to

Multiply by

pascal (Pa) pascal (Pa) pascal (Pa) metre/second2 (m/s2) joule (J) joule (J) 1/metre (1/m) degree Celsius watt (W) watt (W) newton (N) newton-metre (N  m) kilogram (kg) pascal (Pa) pascal (Pa) pascal (Pa) kilogram (kg) metre/second (m/s) newton (N) joule (J) joule (J)

3.376 85 E03 2.490 82 E02 2.488 4 E02 2.540 000*E02 1.000 182 E00 1.000 017 E00 1.000 000*E02 tC  tK  273.15 6.973 333 E01 4.184 000*E03 9.806 650*E00 9.806 650*E00 9.806 650*E00 9.806 650*E04 9.806 650*E00 9.806 650*E06 1.000 000*E00 2.777 778 E01 9.806 650*E00 3.600 000*E06 3.600 655 E06

joule (J)

3.600 061 E06

newton (N) pascal (Pa) metre/second (m/s) candela/metre2 (cd/m2) joule/metre2 (J/m2) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre3 (m3) lumen/metre2 (lm/m2) weber (Wb) siemens (S) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre2 (m2) metre2 (m2) metre/second (m/s) kilometre/hour pascal (Pa) radian (rad) second (s) second (s) second (s) ampere/metre (A/m) ohm ( ) ohm-metre (  m) newton (N) newton-metre (N  m) kilogram (kg) kilogram (kg) kilogram/metre2 (kg/m2) kilogram/metre3 (kg/m3) metre3 (m3) metre3 (m3) metre (m) metre3 (m3) kilogram (kg) kilogram/pascal-secondmetre2 (kg/Pa  s  m2) kilogram/pascal-secondmetre2 (kg/Pa  s  m2)

4.448 222 E03 6.894 757 E06 5.144 444 E01 3.183 099 E03 4.184 000*E04 5.556 000*E03 4.828 041 E03 5.559 552*E03 9.460 54 E15 3.048* E01 2.011 68* E01 1.000 000*E03 1.000 000*E00 1.000 000*E08 1.000 000*E00 2.540 000*E08 1.000 000*E06 2.540 000*E05 1.852 000*E03 1.853 184*E03 1.609 344*E03 1.609 347 E03 2.589 988 E06 2.589 998 E06 4.470 400*E01 1.609 344*E00 1.333 224 E02 2.908 882 E04 6.000 000 E01 5.983 617 E01 2.268 000 E06 7.957 747 E01 1.000 495 E00 1.000 000*E02 2.780 139 E01 7.061 552 E03 2.834 952 E02 3.110 348 E02 3.390 575 E02 1.729 994 E03 2.841 306 E05 2.957 353 E05 3.085 678 E16 8.809 768 E03 1.555 174 E03 5.721 35 E11 5.745 25 E11

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.4

SI Conversion Factors

To convert from perm-inch (08C) perm-inch (238C) phot pica (printer’s) pint (U.S. dry) pint (U.S. liquid) point (printer’s) poise (absolute viscosity) poundal poundal/foot2 poundal-second/foot2 pound-force (lbf avoirdupois) pound-force-inch pound-force-foot pound-force-foot/inch pound-force-inch/inch pound-force/inch pound-force/foot pound-force/foot2 pound-force/inch2 (psi) pound-force-second/foot2 pound-mass (lbm avoirdupois) pound-mass (troy or apothecary) pound-mass-foot2 (moment of inertia) pound-mass-inch2 (moment of inertia) pound-mass/foot2 pound-mass/second pound-mass/minute pound-mass/foot3 pound-mass/inch3 pound-mass/gallon (U.K. liquid) pound-mass/gallon (U.S. liquid) pound-mass/foot-second quart (U.S. dry) quart (U.S. liquid) rad (radiation dose absorbed) rem (dose equivalent) rhe rod (U.S. survey)a roentgen second (angle) second (sidereal) section (U.S. survey)a shake slug slug/foot3 slug/foot-second statampere statcoulomb statfarad stathenry statmho statohm statvolt stere stilb stokes (kinematic viscosity) tablespoon teaspoon ton (assay) ton (long, 2,240 lbm) ton (metric) ton (nuclear equivalent of TNT) ton (register) ton (short, 2,000 lbm) ton (short, mass)/hour ton (long, mass)/yard3 tonne torr (mm Hg, 08C) township (U.S. survey)a unit pole

(Continued ) to

Multiply by

kilogram/pascal-secondmetre (kg/Pa  s  m) kilogram/pascal-secondmetre (kg/Pa  s  m) lumen/metre2 (lm/m2) metre (m) metre3 (m3) metre3 (m3) metre pascal-second (Pa  s) newton (N) pascal (Pa) pascal-second (Pa  s) newton (N) newton-metre (N  m) newton-metre (N  m) newton-metre/metre (N  m/m) newton-metre/metre (N  m/m) newton/metre (N/m) newton/metre (N/m) pascal (Pa) pascal (Pa) pascal-second (Pa  s) kilogram (kg) kilogram (kg) kilogram-metre2 (kg  m2) kilogram-metre2 (kg  m2) kilogram/metre2 (kg/m2) kilogram/second (kg/s) kilogram/second (kg/s) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) pascal-second (Pa  s) metre3 (m3) metre3 (m3) gray (Gy) sievert (Sv) metre2/newton-second (m2/N  s) metre (m) coulomb/kilogram (C/kg) radian (rad) second (s) metre2 (m2) second (s) kilogram (kg) kilogram/metre3 (kg/m3) pascal-second (Pa  s) ampere (A) coulomb (C) farad (F) henry (H) siemens (S) ohm ( ) volt (V) metre3 (m3) candela/metre2 (cd/m2) metre2/second (m2/s) metre3 (m3) metre3 (m3) kilogram (kg) kilogram (kg) kilogram (kg) joule (J) metre3 (m3) kilogram (kg) kilogram/second (kg/s) kilogram/metre3 (kg/m3) kilogram (kg) pascal (Pa) metre2 (m2) weber (Wb)

1.453 22 E12 1.459 29 E12 1.000 000*E04 4.217 518 E03 5.506 105 E04 4.731 765 E04 3.514 598 E04 1.000 000*E01 1.382 550 E01 1.488 164 E00 1.488 164 E00 4.448 222 E00 1.129 848 E01 1.355 818 E00 5.337 866 E01 4.448 222 E00 1.751 268 E02 1.459 390 E01 4.788 026 E01 6.894 757 E03 4.788 026 E01 4.535 924 E01 3.732 417 E01 4.214 011 E02 2.926 397 E04 4.882 428 E00 4.535 924 E01 7.559 873 E03 1.601 846 E01 2.767 990 E04 9.977 637 E01 1.198 264 E02 1.488 164 E00 1.101 221 E03 9.463 529 E04 1.000 000*E02 1.000 000*E02 1.000 000*E01 5.029 210 E00 2.580 000*E04 4.848 137 E06 9.972 696 E01 2.589 998 E06 1.000 000*E08 1.459 390 E01 5.153 788 E02 4.788 026 E01 3.335 641 E10 3.335 641 E10 1.112 650 E12 8.987 552 E11 1.112 650 E12 8.987 552 E11 2.997 925 E02 1.000 000*E00 1.000 000*E04 1.000 000*E04 1.478 676 E05 4.928 922 E06 2.916 667 E02 1.016 047 E03 1.000 000*E03 4.184 000*E09 2.831 685 E00 9.071 847 E02 2.519 958 E01 1.328 939 E03 1.000 000*E03 1.333 22 E02 9.323 994 E07 1.256 637 E07

1-23

1-24

MEASURING UNITS Table 1.2.4

SI Conversion Factors

(Continued )

To convert from

to b

volt, international U.S. (VINTUS) volt, U.S. legal 1948 (VUS48) watt, international U.S. (WINTUS)b watt, U.S. legal 1948 (WUS48) watt/centimetre2 watt-hour watt-second yard yard2 yard3 yard3/minute year (calendar) year (sidereal) year (tropical)

Multiply by

volt (V) volt (V) watt (W) watt (W) watt/metre2 (W/m2) joule (J) joule (J) metre (m) metre2 (m2) metre3 (m3) metre3/second (m3/s) second (s) second (s) second (s)

1.000 338 E00 1.000 008 E00 1.000 182 E00 1.000 017 E00 1.000 000*E04 3.600 000*E03 1.000 000*E00 9.144 000*E01 8.361 274 E01 7.645 549 E01 1.274 258 E02 3.153 600*E07 3.155 815 E07 3.155 693 E07

Based on the U.S. survey foot (1 ft  1,200/3,937 m). b In 1948 a new international agreement was reached on absolute electrical units, which changed the value of the volt used in this country by about 300 parts per million. Again in 1969 a new base of reference was internationally adopted making a further change of 8.4 parts per million. These changes (and also changes in ampere, joule, watt, coulomb) require careful terminology and conversion factors for exact use of old information. Terms used in this guide are: Volt as used prior to January 1948—volt, international U.S. (VINTUS) Volt as used between January 1948 and January 1969—volt, U.S. legal 1948 (VINT48) Volt as used since January 1969—volt (V) Identical treatment is given the ampere, coulomb, watt, and joule. c This value was adopted in 1956. Some of the older International Tables use the value 1.055 04 E03. The exact conversion factor is 1.055 055 852 62*E03. d Moment of inertia of a plane section about a specified axis. e In 1964, the General Conference on Weights and Measures adopted the name “litre” as a special name for the cubic decimetre. Prior to this decision the litre differed slightly (previous value, 1.000028 dm3), and in expression of precision, volume measurement, this fact must be kept in mind. a

SYSTEMS OF UNITS

The principal units of interest to mechanical engineers can be derived from three base units which are considered to be dimensionally independent of each other. The British “gravitational system,” in common use in the United States, uses units of length, force, and time as base units and is also called the “foot-pound-second system.” The metric system, on the other hand, is based on the meter, kilogram, and second, units of length, mass, and time, and is often designated as the “MKS system.” During the nineteenth century a metric “gravitational system,” based on a kilogram-force (also called a “kilopond”) came into general use. With the development of the International System of Units (SI), based as it is on the original metric system for mechanical units, and the general requirements by members of the European Community that only SI units be used, it is anticipated that the kilogram-force will fall into disuse to be replaced by the newton, the SI unit of force. Table 1.2.5 gives the base units of four systems with the corresponding derived unit given in parentheses. In the definitions given below, the “standard kilogram body” refers to the international kilogram prototype, a platinum-iridium cylinder kept in the International Bureau of Weights and Measures in Sèvres, just outside Paris. The “standard pound body” is related to the kilogram by a precise numerical factor: 1 lb  0.453 592 37 kg. This new “unified” pound has replaced the somewhat smaller Imperial pound of the United Kingdom and the slightly larger pound of the United States (see NBS Spec. Pub. 447). The “standard locality” means sea level, 458 latitude,

or more strictly any locality in which the acceleration due to gravity has the value 9.80 665 m/s2  32.1740 ft/s2, which may be called the standard acceleration (Table 1.2.6). The pound force is the force required to support the standard pound body against gravity, in vacuo, in the standard locality; or, it is the force which, if applied to the standard pound body, supposed free to move, would give that body the “standard acceleration.” The word pound is used for the unit of both force and mass and consequently is ambiguous. To avoid uncertainty, it is desirable to call the units “pound force” and “pound mass,” respectively. The slug has been defined as that mass which will accelerate at 1 ft/s2 when acted upon by a one pound force. It is therefore equal to 32.1740 pound-mass. The kilogram force is the force required to support the standard kilogram against gravity, in vacuo, in the standard locality; or, it is the force which, if applied to the standard kilogram body, supposed free to move, would give that body the “standard acceleration.” The word kilogram is used for the unit of both force and mass and consequently is ambiguous. It is for this reason that the General Conference on Weights and Measures declared (in 1901) that the kilogram was the unit of mass, a concept incorporated into SI when it was formally approved in 1960. The dyne is the force which, if applied to the standard gram body, would give that body an acceleration of 1 cm/s2; i.e., 1 dyne  1/980.665 of a gram force. The newton is that force which will impart to a 1-kilogram mass an acceleration of 1 m/s2.

Table 1.2.5

Systems of Units

Quantity

Dimensions of units in terms of L/M/F/T

British “gravitational system”

Metric “gravitational system”

L M F T

1 ft (1 slug) 1 lb 1s

1m

Length Mass Force Time

1 kg 1s

CGS system

SI system

1 cm 1g (1 dyne) 1s

1m 1 kg (1 N) 1s

TIME Table 1.2.6

1-25

Acceleration of Gravity g

Latitude, deg

m/s

ft/s

0 10 20 30 40

9.780 9.782 9.786 9.793 9.802

32.088 32.093 32.108 32.130 32.158

2

2

g

g/g

Latitude, deg

m/s

ft/s2

g/g0

0.9973 0.9975 0.9979 0.9986 0.9995

50 60 70 80 90

9.811 9.819 9.826 9.831 9.832

32.187 32.215 32.238 32.253 32.258

1.0004 1.0013 1.0020 1.0024 1.0026

0

2

NOTE: Correction for altitude above sea level: 3 mm/s2 for each 1,000 m; 0.003 ft/s2 for each 1,000 ft. SOURCE: U.S. Coast and Geodetic Survey, 1912.

TEMPERATURE

The SI unit for thermodynamic temperature is the kelvin, K, which is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Thus 273.16 K is the fixed (base) point on the kelvin scale. Another unit used for the measurement of temperature is degrees Celsius (formerly centigrade), 8C. The relation between a thermodynamic temperature T and a Celsius temperature t is t 5 T 2 273.15 K (the ice point of water) Thus the unit Celsius degree is equal to the unit kelvin, and a difference of temperature would be the same on either scale. In the USCS temperature is measured in degrees Fahrenheit, F. The relation between the Celsius and the Fahrenheit scales is t8C 5 st8F 2 32d/1.8 (For temperature-conversion tables, see Sec. 4.) TERRESTRIAL GRAVITY Standard acceleration of gravity is g 0  9.80665 m per sec per sec,

or 32.1740 ft per sec per sec. This value g0 is assumed to be the value of g at sea level and latitude 458. MOHS SCALE OF HARDNESS

This scale is an arbitrary one which is used to describe the hardness of several mineral substances on a scale of 1 through 10 (Table 1.2.7). The given number indicates a higher relative hardness compared with that of substances below it; and a lower relative hardness than those above it. For example, an unknown substance is scratched by quartz, but it, in turn, scratches feldspar. The unknown has a hardness of between 6 and 7 on the Mohs scale. Table 1.2.7 1. 2. 3. 4.

Talc Gypsum Calc-spar Fluorspar

Mohs Scale of Hardness 5. Apatite 6. Feldspar 7. Quartz

8. Topaz 9. Sapphire 10. Diamond

TIME Kinds of Time Three kinds of time are recognized by astronomers: sidereal, apparent solar, and mean solar time. The sidereal day is the interval between two consecutive transits of some fixed celestial object across any given meridian, or it is the interval required by the earth to make one complete revolution on its axis. The interval is constant, but it is inconvenient as a time unit because the noon of the sidereal day occurs at all hours of the day and night. The apparent solar day is the interval between two consecutive transits of the sun across any given meridian. On account of the variable distance between the sun and earth, the variable speed of the earth in its orbit, the effect of the moon, etc., this interval is not constant and consequently cannot be kept by any simple mechanisms, such as clocks or watches. To overcome the objection noted above, the mean solar day was devised. The mean solar day is

the length of the average apparent solar day. Like the sidereal day it is constant, and like the apparent solar day its noon always occurs at approximately the same time of day. By international agreement, beginning Jan. 1, 1925, the astronomical day, like the civil day, is from midnight to midnight. The hours of the astronomical day run from 0 to 24, and the hours of the civil day usually run from 0 to 12 A.M. and 0 to 12 P.M. In some countries the hours of the civil day also run from 0 to 24. The Year Three different kinds of year are used: the sidereal, the tropical, and the anomalistic. The sidereal year is the time taken by the earth to complete one revolution around the sun from a given star to the same star again. Its length is 365 days, 6 hours, 9 minutes, and 9 seconds. The tropical year is the time included between two successive passages of the vernal equinox by the sun, and since the equinox moves westward 50.2 seconds of arc a year, the tropical year is shorter by 20 minutes 23 seconds in time than the sidereal year. As the seasons depend upon the earth’s position with respect to the equinox, the tropical year is the year of civil reckoning. The anomalistic year is the interval between two successive passages of the perihelion, viz., the time of the earth’s nearest approach to the sun. The anomalistic year is used only in special calculations in astronomy. The Second Although the second is ordinarily defined as 1/86,400 of the mean solar day, this is not sufficiently precise for many scientific purposes. Scientists have adopted more precise definitions for specific purposes: in 1956, one in terms of the length of the tropical year 1900 and, more recently, in 1967, one in terms of a specific atomic frequency. Frequency is the reciprocal of time for 1 cycle; the unit of frequency is the hertz (Hz), defined as 1 cycle/s. The Calendar The Gregorian calendar, now used in most of the civilized world, was adopted in Catholic countries of Europe in 1582 and in Great Britain and her colonies Jan. 1, 1752. The average length of the Gregorian calendar year is 365 1⁄4 2 3⁄400 days, or 365.2425 days. This is equivalent to 365 days, 5 hours, 49 minutes, 12 seconds. The length of the tropical year is 365.2422 days, or 365 days, 5 hours, 48 minutes, 46 seconds. Thus the Gregorian calendar year is longer than the tropical year by 0.0003 day, or 26 seconds. This difference amounts to 1 day in slightly more than 3,300 years and can properly be neglected. Standard Time Prior to 1883, each city of the United States had its own time, which was determined by the time of passage of the sun across the local meridian. A system of standard time had been used since its first adoption by the railroads in 1883 but was first legalized on Mar. 19, 1918, when Congress directed the Interstate Commerce Commission to establish limits of the standard time zones. Congress took no further steps until the Uniform Time Act of 1966 was enacted, followed with an amendment in 1972. This legislation, referred to as “the Act,” transferred the regulation and enforcement of the law to the Department of Transportation. By the legislation of 1918, with some modifications by the Act, the contiguous United States is divided into four time zones, each of which, theoretically, was to span 15 degrees of longitude. The first, the Eastern zone, extends from the Atlantic westward to include most of Michigan and Indiana, the eastern parts of Kentucky and Tennessee, Georgia, and Florida, except the west half of the panhandle. Eastern standard time is

1-26

MEASURING UNITS

based upon the mean solar time of the 75th meridian west of Greenwich, and is 5 hours slower than Greenwich Mean Time (GMT). (See also discussion of UTC below.) The second or Central zone extends westward to include most of North Dakota, about half of South Dakota and Nebraska, most of Kansas, Oklahoma, and all but the two most westerly counties of Texas. Central standard time is based upon the mean solar time of the 90th meridian west of Greenwich, and is 6 hours slower than GMT. The third or Mountain zone extends westward to include Montana, most of Idaho, one county of Oregon, Utah, and Arizona. Mountain standard time is based upon the mean solar time of the 105th meridian west of Greenwich, and is 7 hours slower than GMT. The fourth or Pacific zone includes all of the remaining 48 contiguous states. Pacific standard time is based on the mean solar time of the 120th meridian west of Greenwich, and is 8 hours slower than GMT. Exact locations of boundaries may be obtained from the Department of Transportation. In addition to the above four zones there are four others that apply to the noncontiguous states and islands. The most easterly is the Atlantic zone, which includes Puerto Rico and the Virgin Islands, where the time is 4 hours slower than GMT. Eastern standard time is used in the Panama Canal strip. To the west of the Pacific time zone there are the Yukon, the Alaska-Hawaii, and Bering zones where the times are, respectively, 9, 10, and 11 hours slower than GMT. The system of standard time has been adopted in all civilized countries and is used by ships on the high seas. The Act directs that from the first Sunday in April to the last Sunday in October, the time in each zone is to be advanced one hour for advanced time or daylight saving time (DST). However, any state-bystate enactment may exempt the entire state from using advanced time. By this provision Arizona and Hawaii do not observe advanced time (as of 1973). By the 1972 amendment to the Act, a state split by a timezone boundary may exempt from using advanced time all that part which is in one zone without affecting the rest of the state. By this amendment, 80 counties of Indiana in the Eastern zone are exempt from using advanced time, while 6 counties in the northwest corner and 6 counties in the southwest, which are in Central zone, do observe advanced time. Pursuant to its assignment of carrying out the Act, the Department of Transportation has stipulated that municipalities located on the boundary between the Eastern and Central zones are in the Central zone; those on the boundary between the Central and Mountain zones are in the Mountain zone (except that Murdo, SD, is in the Central zone); those on the boundary between Mountain and Pacific time zones are in the Mountain zone. In such places, when the time is given, it should be specified as Central, Mountain, etc. Standard Time Signals The National Institute of Standards and Technology broadcasts time signals from station WWV, Ft. Collins, CO, and from station WWVH, near Kekaha, Kaui, HI. The broadcasts by WWV are on radio carrier frequencies of 2.5, 5, 10, 15, and 20 MHz, while those by WWVH are on radio carrier frequencies of 2.5, 5, 10, and 15 MHz. Effective Jan. 1, 1975, time announcements by both WWV and WWVH are referred to as Coordinated Universal Time, UTC, the international coordinated time scale used around the world for most timekeeping purposes. UTC is generated by reference to International Atomic Time (TAI), which is determined by the Bureau International de l’Heure on the basis of atomic clocks operating in various establishments in accordance with the definition of the second. Since the difference between UTC and TAI is defined to be a whole number of seconds, a “leap second” is periodically added to or subtracted from UTC to take into account variations in the rotation of the earth. Time (i.e., clock time) is given in terms of 0 to 24 hours a day, starting with 0000 at midnight at Greenwich zero longitude. The beginning of each 0.8-second-long audio tone marks the end of an announced time interval. For example, at 2:15 P.M., UTC, the voice announcement would be: “At the tone fourteen hours fifteen minutes Coordinated Universal Time,” given during the last 7.5 seconds of each minute. The tone markers from both stations are given simultaneously, but owing to propagation interferences may not be received simultaneously.

Beginning 1 minute after the hour, a 600-Hz signal is broadcast for about 45 s. At 2 min after the hour, the standard musical pitch of 440 Hz is broadcast for about 45 s. For the remaining 57 min of the hour, alternating tones of 600 and 500 Hz are broadcast for the first 45 s of each minute (see NIST Spec. Pub. 432). The time signal can also be received via long-distance telephone service from Ft. Collins. In addition to providing the musical pitch, these tone signals may be of use as markers for automated recorders and other such devices. DENSITY AND RELATIVE DENSITY Density of a body is its mass per unit volume. With SI units densities

are in kilograms per cubic meter. However, giving densities in grams per cubic centimeter has been common. With the USCS, densities are given in pounds per mass cubic foot. Table 1.2.8 Relative Densities at 608/608F Corresponding to Degrees API and Weights per U.S. Gallon at 608F 141.5 ¢Calculated from the formula, relative density 5 ≤ 131.5 1 deg API Degrees API

Relative density

Lb per U.S. gallon

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

1.0000 0.9930 0.9861 0.9792 0.9725 0.9659 0.9593 0.9529 0.9465 0.9402 0.9340 0.9279 0.9218 0.9159 0.9100 0.9042 0.8984 0.8927 0.8871 0.8816 0.8762 0.8708 0.8654 0.8602 0.8550 0.8498 0.8448 0.8398 0.8348 0.8299 0.8251 0.8203 0.8155 0.8109 0.8063 0.8017 0.7972 0.7927 0.7883 0.7839 0.7796 0.7753 0.7711 0.7669 0.7628 0.7587

8.328 8.270 8.212 8.155 8.099 8.044 7.989 7.935 7.882 7.830 7.778 7.727 7.676 7.627 7.578 7.529 7.481 7.434 7.387 7.341 7.296 7.251 7.206 7.163 7.119 7.076 7.034 6.993 6.951 6.910 6.870 6.830 6.790 6.752 6.713 6.675 6.637 6.600 6.563 6.526 6.490 6.455 6.420 6.385 6.350 6.316

Degrees API

Relative density

Lb per U.S. gallon

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.7547 0.7507 0.7467 0.7428 0.7389 0.7351 0.7313 0.7275 0.7238 0.7201 0.7165 0.7128 0.7093 0.7057 0.7022 0.6988 0.6953 0.6919 0.6886 0.6852 0.6819 0.6787 0.6754 0.6722 0.6690 0.6659 0.6628 0.6597 0.6566 0.6536 0.6506 0.6476 0.6446 0.6417 0.6388 0.6360 0.6331 0.6303 0.6275 0.6247 0.6220 0.6193 0.6166 0.6139 0.6112

6.283 6.249 6.216 6.184 6.151 6.119 6.087 6.056 6.025 5.994 5.964 5.934 5.904 5.874 5.845 5.817 5.788 5.759 5.731 5.703 5.676 5.649 5.622 5.595 5.568 5.542 5.516 5.491 5.465 5.440 5.415 5.390 5.365 5.341 5.316 5.293 5.269 5.246 5.222 5.199 5.176 5.154 5.131 5.109 5.086

NOTE: The weights in this table are weights in air at 608F with humidity 50 percent and pressure 760 mm.

CONVERSION AND EQUIVALENCY TABLES Table 1.2.9 Relative Densities at 608/608F Corresponding to Degrees Baumé for Liquids Lighter than Water and Weights per U.S. Gallon at 608F 608 140 ¢Calculated from the formula, relative density F5 ≤ 608 130 1 deg Baumé Degrees Baumé

Relative density

Lb per gallon

Degrees Baumé

Relative density

Lb per gallon

10.0 11.0 12.0 13.0

1.0000 0.9929 0.9859 0.9790

8.328 8.269 8.211 8.153

56.0 57.0 58.0 59.0

0.7527 0.7487 0.7447 0.7407

6.266 6.233 6.199 6.166

14.0 15.0 16.0 17.0

0.9722 0.9655 0.9589 0.9524

8.096 8.041 7.986 7.931

60.0 61.0 62.0 63.0

0.7368 0.7330 0.7292 0.7254

6.134 6.102 6.070 6.038

18.0 19.0 20.0 21.0

0.9459 0.9396 0.9333 0.9272

7.877 7.825 7.772 7.721

64.0 65.0 66.0 67.0

0.7216 0.7179 0.7143 0.7107

6.007 5.976 5.946 5.916

22.0 23.0 24.0 25.0

0.9211 0.9150 0.9091 0.9032

7.670 7.620 7.570 7.522

68.0 69.0 70.0 71.0

0.7071 0.7035 0.7000 0.6965

5.886 5.856 5.827 5.798

26.0 27.0 28.0 29.0

0.8974 0.8917 0.8861 0.8805

7.473 7.425 7.378 7.332

72.0 73.0 74.0 75.0

0.6931 0.6897 0.6863 0.6829

5.769 5.741 5.712 5.685

30.0 31.0 32.0 33.0

0.8750 0.8696 0.8642 0.8589

7.286 7.241 7.196 7.152

76.0 77.0 78.0 79.0

0.6796 0.6763 0.6731 0.6699

5.657 5.629 5.602 5.576

34.0 35.0 36.0 37.0

0.8537 0.8485 0.8434 0.8383

7.108 7.065 7.022 6.980

80.0 81.0 82.0 83.0

0.6667 0.6635 0.6604 0.6573

5.549 5.522 5.497 5.471

38.0 39.0 40.0 41.0

0.8333 0.8284 0.8235 0.8187

6.939 6.898 6.857 6.817

84.0 85.0 86.0 87.0

0.6542 0.6512 0.6482 0.6452

5.445 5.420 5.395 5.370

42.0 43.0 44.0 45.0

0.8140 0.8092 0.8046 0.8000

6.777 6.738 6.699 6.661

88.0 89.0 90.0 91.0

0.6422 0.6393 0.6364 0.6335

5.345 5.320 5.296 5.272

46.0 47.0 48.0 49.0

0.7955 0.7910 0.7865 0.7821

6.623 6.586 6.548 6.511

92.0 93.0 94.0 95.0

0.6306 0.6278 0.6250 0.6222

5.248 5.225 5.201 5.178

50.0 51.0 52.0 53.0 54.0 55.0

0.7778 0.7735 0.7692 0.7650 0.7609 0.7568

6.476 6.440 6.404 6.369 6.334 6.300

96.0 97.0 98.0 99.0 100.0

0.6195 0.6167 0.6140 0.6114 0.6087

5.155 5.132 5.100 5.088 5.066

1-27

equal volumes, the ratio of the molecular weight of the gas to that of air may be used as the relative density of the gas. When this is done, the molecular weight of air may be taken as 28.9644. The relative density of liquids is usually measured by means of a hydrometer. In addition to a scale reading in relative density as defined above, other arbitrary scales for hydrometers are used in various trades and industries. The most common of these are the API and Baumé. The API (American Petroleum Institute) scale is approved by the American Petroleum Institute, the ASTM, the U.S. Bureau of Mines, and the National Bureau of Standards and is recommended for exclusive use in the U.S. petroleum industry, superseding the Baumé scale for liquids lighter than water. The relation between API degrees and relative density (see Table 1.2.8) is expressed by the following equation: Degrees API 5

141.5 2 131.5 rel dens 608/608F

The relative densities corresponding to the indications of the Baumé hydrometer are given in Tables 1.2.9 and 1.2.10. Table 1.2.10 Relative Densities at 608/608F Corresponding to Degrees Baumé for Liquids Heavier than Water 608 145 ¢Calculated from the formula, relative density F5 ≤ 608 145 2 deg Baumé Degrees Baumé

Relative density

Degrees Baumé

Relative density

Degrees Baumé

Relative density

0 1 2 3

1.0000 1.0069 1.0140 1.0211

24 25 26 27

1.1983 1.2083 1.2185 1.2288

48 49 50 51

1.4948 1.5104 1.5263 1.5426

4 5 6 7

1.0284 1.0357 1.0432 1.0507

28 29 30 31

1.2393 1.2500 1.2609 1.2719

52 53 54 55

1.5591 1.5761 1.5934 1.6111

8 9 10 11

1.0584 1.0662 1.0741 1.0821

32 33 34 35

1.2832 1.2946 1.3063 1.3182

56 57 58 59

1.6292 1.6477 1.6667 1.6860

12 13 14 15

1.0902 1.0985 1.1069 1.1154

36 37 38 39

1.3303 1.3426 1.3551 1.3679

60 61 62 63

1.7059 1.7262 1.7470 1.7683

16 17 18 19

1.1240 1.1328 1.1417 1.1508

40 41 42 43

1.3810 1.3942 1.4078 1.4216

64 65 66 67

1.7901 1.8125 1.8354 1.8590

20 21 22 23

1.1600 1.1694 1.1789 1.1885

44 45 46 47

1.4356 1.4500 1.4646 1.4796

68 69 70

1.8831 1.9079 1.9333

CONVERSION AND EQUIVALENCY TABLES Note for Use of Conversion Tables (Tables 1.2.11 through 1.2.34)

Relative density is the ratio of the density of one substance to that of a

second (or reference) substance, both at some specified temperature. Use of the earlier term specific gravity for this quantity is discouraged. For solids and liquids water is almost universally used as the reference substance. Physicists use a reference temperature of 48C ( 39.28F); U.S. engineers commonly use 608F. With the introduction of SI units, it may be found desirable to use 598F, since 598F and 158C are equivalents. For gases, relative density is generally the ratio of the density of the gas to that of air, both at the same temperature, pressure, and dryness (as regards water vapor). Because equal numbers of moles of gases occupy

Subscripts after any figure, 0s, 9s, etc., mean that that figure is to be repeated the indicated number of times.

1-28

MEASURING UNITS

Table 1.2.11 Centimetres 1 2.540 30.48 91.44 100 2012 100000 160934

Length Equivalents Inches 0.3937 1 12 36 39.37 792 39370 63360

Feet 0.03281 0.08333 1 3 3.281 66 3281 5280

Yards 0.01094 0.02778 0.3333 1 1.0936 22 1093.6 1760

Metres 0.01 0.0254 0.3048 0.9144 1 20.12 1000 1609

Chains

Kilometres

0.034971 0.001263 0.01515 0.04545 0.04971 1 49.71 80

5

10 0.04254 0.033048 0.039144 0.001 0.02012 1 1.609

Miles 0.056214 0.041578 0.031894 0.035682 0.036214 0.0125 0.6214 1

(As used by metrology laboratories for precise measurements, including measurements of surface texture)*

Angstrom units Å

Surface texture (U.S.), microinch min

Light bands,† monochromatic helium light count ‡

Surface texture foreign, mm

Precision measurements, § 0.0001 in

Close-tolerance measurements, 0.001 in (mils)

Metric unit, mm

USCS unit, in

1 254 2937.5 10,000 25,400 254,000 10,000,000 254,000,000

0.003937 1 11.566 39.37 100 1000 39,370 1,000,000

0.0003404 0.086 1 3.404 8.646 86.46 3404 86,460

0.0001 0.0254 0.29375 1 2.54 25.4 1000 25,400

0.043937 0.01 0.11566 0.3937 1 10 393.7 10,000

0.053937 0.001 0.011566 0.03937 0.1 1 39.37 1000

0.061 0.04254 0.0329375 0.001 0.00254 0.0254 1 25.4

0.083937 0.051 0.0411566 0.043937 0.0001 0.001 0.03937 1

* Computed by J. A. Broadston. † One light band equals one-half corresponding wavelength. Visible-light wavelengths range from red at 6,500 Å to violet at 4,100 Å. ‡ One helium light band  0.000011661 in  2937.5 Å; one krypton 86 light band  0.0000119 in  3,022.5 Å; one mercury 198 light band  0.00001075 in  2,730 Å. § The designations “precision measurements,” etc., are not necessarily used in all metrology laboratories.

CONVERSION AND EQUIVALENCY TABLES Table 1.2.12

1-29

Conversion of Lengths*

Inches to millimetres

Millimetres to inches

Feet to metres

Metres to feet

Yards to metres

Metres to yards

Miles to kilometres

Kilometres to miles

1 2 3 4

25.40 50.80 76.20 101.60

0.03937 0.07874 0.1181 0.1575

0.3048 0.6096 0.9144 1.219

3.281 6.562 9.843 13.12

0.9144 1.829 2.743 3.658

1.094 2.187 3.281 4.374

1.609 3.219 4.828 6.437

0.6214 1.243 1.864 2.485

5 6 7 8 9

127.00 152.40 177.80 203.20 228.60

0.1969 0.2362 0.2756 0.3150 0.3543

1.524 1.829 2.134 2.438 2.743

16.40 19.69 22.97 26.25 29.53

4.572 5.486 6.401 7.315 8.230

5.468 6.562 7.655 8.749 9.843

6.047 9.656 11.27 12.87 14.48

3.107 3.728 4.350 4.971 5.592

* EXAMPLE: 1 in  25.40 mm.

Common fractions of an inch to millimetres (from 1⁄64 to 1 in) 64ths

Millimetres

64ths

Millimetres

64th

Millimetres

64ths

Millimetres

64ths

Millimetres

64ths

Millimetres

1 2 3 4

0.397 0.794 1.191 1.588

13 14 15 16

5.159 5.556 5.953 6.350

25 26 27 28

9.922 10.319 10.716 11.112

37 38 39 40

14.684 15.081 15.478 15.875

49 50 51 52

19.447 19.844 20.241 20.638

57 58 59 60

22.622 23.019 23.416 23.812

5 6 7 8

1.984 2.381 2.778 3.175

17 18 19 20

6.747 7.144 7.541 7.938

29 30 31 32

11.509 11.906 12.303 12.700

41 42 43 44

16.272 16.669 17.066 17.462

53 54 55 56

21.034. 21.431 21.828 22.225

61 62 63 64

24.209 24.606 25.003 25.400

9 10 11 12

3.572 3.969 4.366 4.762

21 22 23 24

8.334 8.731 9.128 9.525

33 34 35 36

13.097 13.494 13.891 14.288

45 46 47 48

17.859 18.256 18.653 19.050

0

1

2

3

4

5

6

7

8

9

.0 .1 .2 .3 .4

2.540 5.080 7.620 10.160

0.254 2.794 5.334 7.874 10.414

0.508 3.048 5.588 8.128 10.668

0.762 3.302 5.842 8.382 10.922

1.016 3.556 6.096 8.636 11.176

1.270 3.810 6.350 8.890 11.430

1.524 4.064 6.604 9.144 11.684

1.778 4.318 6.858 9.398 11.938

2.032 4.572 7.112 9.652 12.192

2.286 4.826 7.366 9.906 12.446

.5 .6 .7 .8 .9

12.700 15.240 17.780 20.320 22.860

12.954 15.494 18.034 20.574 23.114

13.208 15.748 18.288 20.828 23.368

13.462 16.002 18.542 21.082 23.622

13.716 16.256 18.796 21.336 23.876

13.970 16.510 19.050 21.590 24.130

14.224 16.764 19.304 21.844 24.384

14.478 17.018 19.558 22.098 24.638

14.732 17.272 19.812 22.352 24.892

14.986 17.526 20.066 22.606 25.146

0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

0 1 2 3 4

0.3937 0.7874 1.1811 1.5748

0.0394 0.4331 0.8268 1.2205 1.6142

0.0787 0.4724 0.8661 1.2598 1.6535

0.1181 0.5118 0.9055 1.2992 1.6929

0.1575 0.5512 0.9449 1.3386 1.7323

0.1969 0.5906 0.9843 1.3780 1.7717

0.2362 0.6299 1.0236 1.4173 1.8110

0.2756 0.6693 1.0630 1.4567 1.8504

0.3150 0.7087 1.1024 1.4961 1.8898

0.3543 0.7480 1.1417 1.5354 1.9291

5 6 7 8 9

1.9685 2.3622 2.7559 3.1496 3.5433

2.0079 2.4016 2.7953 3.1890 3.5827

2.0472 2.4409 2.8346 3.2283 3.6220

2.0866 2.4803 2.8740 3.2677 3.6614

2.1260 2.5197 2.9134 3.3071 3.7008

2.1654 2.5591 2.9528 3.3465 3.7402

2.2047 2.5984 2.9921 3.3858 3.7795

2.2441 2.6378 3.0315 3.4252 3.8189

2.2835 2.6772 3.0709 3.4646 3.8583

2.3228 2.7165 3.1102 3.5039 3.8976

Decimals of an inch to millimetres (0.01 to 0.99 in)

Millimetres to decimals of an inch (from 1 to 99 mm)

1-30

MEASURING UNITS

Table 1.2.13 Area Equivalents (1 hectare  100 ares  10,000 centiares or square metres) Square metres

Square inches

Square feet

Square yards

Square rods

Square chains

Roods

Acres

Square miles or sections

1 0.036452 0.09290 0.8361 25.29 404.7 1012 4047 2589988

1550 1 144 1296 39204 627264 1568160 6272640

10.76 0.006944 1 9 272.25 4356 10890 43560 27878400

1.196 0.037716 0.1111 1 30.25 484 1210 4840 3097600

0.0395 0.042551 0.003673 0.03306 1 16 40 160 102400

0.002471 0.051594 0.032296 0.002066 0.0625 1 2.5 10 6400

0.039884 0.066377 0.049183 0.038264 0.02500 0.4 1 4 2560

0.032471 0.061594 0.042296 0.0002066 0.00625 0.1 0.25 1 640

0.063861 0.092491 0.073587 0.063228 0.059766 0.0001562 0.033906 0.001562 1

Table 1.2.14

Conversion of Areas*

Sq in to sq cm

Sq cm to sq in

Sq ft to sq m

Sq m to sq ft

Sq yd to sq m

Sq m to sq yd

Acres to hectares

Hectares to acres

Sq mi to sq km

Sq km to sq mi

1 2 3 4

6.452 12.90 19.35 25.81

0.1550 0.3100 0.4650 0.6200

0.0929 0.1858 0.2787 0.3716

10.76 21.53 32.29: 43.06

0.8361 1.672 2.508 3.345

1.196 2.392 3.588 4.784

0.4047 0.8094 1.214 1.619

2.471 4.942 7.413 9.884

2.590 5.180 7.770 10.360

0.3861 0.7722 1.158 1.544

5 6 7 8 9

32.26 38.71 45.16 51.61 58.06

0.7750 0.9300 1.085 1.240 1.395

0.4645 0.5574 0.6503 0.7432 0.8361

53.82 64.58 75.35 86.11 96.88

4.181 5.017 5.853 6.689 7.525

5.980 7.176 8.372 9.568 10.764

2.023 2.428 2.833 3.237 3.642

12.355 14.826 17.297 19.768 22.239

12.950 15.540 18.130 20.720 23.310

1.931 2.317 2.703 3.089 3.475

* EXAMPLE: 1 in2  6.452 cm2.

Table 1.2.15

Volume and Capacity Equivalents

Cubic inches

Cubic feet

Cubic yards

U.S. Apothecary fluid ounces

Liquid

Dry

U.S. gallons

U.S. bushels

Cubic decimetres or litres

1 1728 46656 1.805 57.75 67.20 231 2150 61.02

0.035787 1 27 0.001044 0.03342 0.03889 0.1337 1.244 0.03531

0.042143 0.03704 1 0.043868 0.001238 0.001440 0.004951 0.04609 0.001308

0.5541 957.5 25853 1 32 37.24 128 1192 33.81

0.01732 29.92 807.9 0.03125 1 1.164 4 37.24 1.057

0.01488 25.71 694.3 0.02686 0.8594 1 3.437 32 0.9081

0.024329 7.481 202.2 0.007812 0.25 0.2909 1 9.309 0.2642

0.034650 0.8036 21.70 0.038392 0.02686 0.03125 0.1074 1 0.02838

0.01639 28.32 764.6 0.02957 0.9464 1.101 3.785 35.24 1

Table 1.2.16

U.S. quarts

Conversion of Volumes or Cubic Measure*

Cu in to mL

mL to cu in

Cu ft to cu m

Cu m to cu ft

Cu yd to cu m

Cu m to cu yd

Gallons to cu ft

Cu ft to gallons

1 2 3 4

16.39 32.77 49.16 65.55

0.06102 0.1220 0.1831 0.2441

0.02832 0.05663 0.08495 0.1133

35.31 70.63 105.9 141.3

0.7646 1.529 2.294 3.058

1.308 2.616 3.924 5.232

0.1337 0.2674 0.4010 0.5347

7.481 14.96 22.44 29.92

5 6 7 8 9

81.94 98.32 114.7 131.1 147.5

0.3051 0.3661 0.4272 0.4882 0.5492

0.1416 0.1699 0.1982 0.2265 0.2549

176.6 211.9 247.2 282.5 317.8

3.823 4.587 5.352 6.116 6.881

6.540 7.848 9.156 10.46 11.77

0.6684 0.8021 0.9358 1.069 1.203

37.40 44.88 52.36 59.84 67.32

* EXAMPLE: 1 in3  16.39 mL.

CONVERSION AND EQUIVALENCY TABLES Table 1.2.17

1-31

Conversion of Volumes or Capacities*

Fluid ounces to mL

mL to fluid ounces

Liquid pints to litres

Litres to liquid pints

Liquid quarts to litres

Litres to liquid quarts

Gallons to litres

Litres to gallons

Bushels to hectolitres

Hectolitres to bushels

1 2 3 4

29.57 59.15 88.72 118.3

0.03381 0.06763 0.1014 0.1353

0.4732 0.9463 1.420 1.893

2.113 4.227 6.340 8.454

0.9463 1.893 2.839 3.785

1.057 2.113 3.170 4.227

3.785 7.571 11.36 15.14

0.2642 0.5284 0.7925 1.057

0.3524 0.7048 1.057 1.410

2.838 5.676 8.513 11.35

5 6 7 8 9

147.9 177.4 207.0 236.6 266.2

0.1691 0.2092 0.2367 0.2705 0.3043

2.366 2.839 3.312 3.785 4.259

4.732 5.678 6.624 7.571 8.517

5.284 6.340 7.397 8.454 9.510

18.93 22.71 26.50 30.28 34.07

1.321 1.585 1.849 2.113 2.378

1.762 2.114 2.467 2.819 3.171

14.19 17.03 19.86 22.70 25.54

10.57 12.68 14.79 16.91 19.02

* EXAMPLE: 1 fluid oz  29.57 mL.

Table 1.2.18

Mass Equivalents Ounces

Pounds

Tons

Kilograms

Grains

Troy and apoth

Avoirdupois

Troy and apoth

Avoirdupois

Short

Long

Metric

1 0.046480 0.03110 0.02835 0.3732 0.4536 907.2 1016 1000

15432 1 480 437.5 5760 7000 1406 156804 15432356

32.15 0.022083 1 0.9115 12 14.58 29167 32667 32151

35.27 0.022286 1.09714 1 13.17 16 3203 35840 35274

2.6792 0.031736 0.08333 0.07595 1 1.215 2431 2722 2679

2.205 0.031429 0.06857 0.0625 0.8229 1 2000 2240 2205

0.021102 0.077143 0.043429 0.043125 0.034114 0.0005 1 1.12 1.102

0.039842 0.076378 0.043061 0.042790 0.033673 0.034464 0.8929 1 0.9842

0.001 0.076480 0.043110 0.042835 0.033732 0.034536 0.9072 1.016 1

Table 1.2.19

Conversion of Masses*

Grams to ounces (avdp)

Pounds (avdp) to kilograms

Kilograms to pounds (avdp)

Short tons (2000 lb) to metric tons

2.205 4.409 6.614 8.818

0.907 1.814 2.722 3.629

1.102 2.205 3.307 4.409

1.016 2.032 3.048 4.064

0.984 1.968 2.953 3.937

4.536 5.443 6.350 7.257 8.165

5.512 6.614 7.716 8.818 9.921

5.080 6.096 7.112 8.128 9.144

4.921 5.905 6.889 7.874 8.858

Grains to grams

Grams to grains

Ounces (avdp) to grams

1 2 3 4

0.06480 0.1296 0.1944 0.2592

15.43 30.86 46.30 61.73

28.35 56.70 85.05 113.40

0.03527 0.07055 0.1058 0.1411

0.4536 0.9072 1.361 1.814

5 6 7 8 9

0.3240 0.3888 0.4536 0.5184 0.5832

77.16 92.59 108.03 123.46 138.89

141.75 170.10 198.45 226.80 255.15

0.1764 0.2116 0.2469 0.2822 0.3175

2.268 2.722 3.175 3.629 4.082

11.02 13.23 15.43 17.64 19.84

Metric tons (1000 kg) to short tons

Long tons (2240 lb) to metric tons

Metric tons to long tons

* EXAMPLE: 1 grain  0.06480 grams.

Table 1.2.20

Pressure Equivalents Columns of mercury at temperature 08C and g  9.80665 m/s2

Columns of water at temperature 158C and g  9.80665 m/s2

Pascals N/m2

Bars 105 N/m2

Poundsf per in2

Atmospheres

cm

in

cm

in

1 100000 6894.8 101326 1333 3386 97.98 248.9

105 1 0.068948 1.0132 0.0133 0.03386 0.0009798 0.002489

0.000145 14.504 1 14.696 0.1934 0.4912 0.01421 0.03609

0.00001 0.9869 0.06805 1 0.01316 0.03342 0.000967 0.002456

0.00075 75.01 5.171 76.000 1 2.540 0.07349 0.1867

0.000295 29.53 2.036 29.92 0.3937 1 0.02893 0.07349

0.01021 1020.7 70.37 1034 13.61 34.56 1 2.540

0.00402 401.8 27.703 407.1 5.357 13.61 0.3937 1

1-32

MEASURING UNITS Table 1.2.21

Conversion of Pressures*

Lb/in2 to bars

Bars to lb/in2

Lb/in2 to atmospheres

Atmospheres to lb/in2

Bars to atmospheres

Atmospheres to bars

1 2 3 4

0.06895 0.13790 0.20684 0.27579

14.504 29.008 43.511 58.015

0.06805 0.13609 0.20414 0.27218

14.696 29.392 44.098 58.784

0.98692 1.9738 2.9607 3.9477

1.01325 2.0265 3.0397 4.0530

5 6 7 8 9

0.34474 0.41368 0.48263 0.55158 0.62053

72.519 87.023 101.53 116.03 130.53

0.34823 0.40826 0.47632 0.54436 0.61241

73.480 88.176 102.87 117.57 132.26

4.9346 5.9215 6.9085 7.8954 8.8823

5.0663 6.0795 7.0927 8.1060 9.1192

* EXAMPLE: 1 lb/in2  0.06895 bar.

Table 1.2.22

Velocity Equivalents

cm/s

m/s

m/min

km/h

ft/s

ft/min

mi/h

Knots

1 100 1.667 27.78 30.48 0.5080 44.70 51.44

0.01 1 0.01667 0.2778 0.3048 0.005080 0.4470 0.5144

0.6 60 1 16.67 18.29 0.3048 26.82 30.87

0.036 3.6 0.06 1 1.097 0.01829 1.609 1.852

0.03281 3.281 0.05468 0.9113 1 0.01667 1.467 1.688

1.9685 196.85 3.281 54.68 60 1 88 101.3

0.02237 2.237 0.03728 0.6214 0.6818 0.01136 1 1.151

0.01944 1.944 0.03240 0.53996 0.59248 0.00987 0.86898 1

Table 1.2.23

Conversion of Linear and Angular Velocities*

cm/s to ft/min

ft/min to cm/s

cm/s to mi/h

mi/h to cm/s

ft/s to mi/h

mi/h to ft/s

rad/s to r/min

r/min to rad/s

1 2 3 4

1.97 3.94 5.91 7.87

0.508 1.016 1.524 2.032

0.0224 0.0447 0.0671 0.0895

44.70 89.41 134.1 178.8

0.682 1.364 2.045 2.727

1.47 2.93 4.40 5.87

9.55 19.10 28.65 38.20

0.1047 0.2094 0.3142 0.4189

5 6 7 8 9

9.84 11.81 13.78 15.75 17.72

2.540 3.048 3.556 4.064 4.572

0.1118 0.1342 0.1566 0.1790 0.2013

223.5 268.2 312.9 357.6 402.3

3.409 4.091 4.773 5.455 6.136

7.33 8.80 10.27 11.73 13.20

47.75 57.30 66.84 76.39 85.94

0.5236 0.6283 0.7330 0.8378 0.9425

* EXAMPLE: 1 cm/s  1.97 ft/min.

Table 1.2.24

Acceleration Equivalents

cm/s

m/s2

m/(h  s)

km/(h  s)

ft/(h  s)

ft/s2

ft/min2

mi/(h  s)

knots/s

1 100 0.02778 27.78 0.008467 30.48 0.008467 44.70 51.44

0.01 1 0.0002778 0.2778 0.00008467 0.3048 0.00008467 0.4470 0.5144

36.00 3600 1 1000 0.3048 1097 0.3048 1609 1852

0.036 3.6 0.001 1 0.0003048 1.097 0.0003048 1.609 1.852

118.1 11811 3.281 3281 1 3600 1 5280 6076

0.03281 3.281 0.0009113 0.9113 0.0002778 1 0.0002778 1.467 1.688

118.1 11811 3.281 3281 1 3600 1 5280 6076

0.02237 2.237 0.0006214 0.6214 0.0001894 0.6818 0.0001894 1 1.151

0.01944 1.944 0.0005400 0.5400 0.0001646 0.4572 0.0001646 0.8690 1

2

CONVERSION AND EQUIVALENCY TABLES Table 1.2.25

1-33

Conversion of Accelerations*

cm/s2 to ft/min2

km/(h  s) to mi/(h  s)

km/(h  s) to knots/s

ft/s2 to mi/(h  s)

ft/s2 to knots/s

ft/min2 to cm/s2

mi/(h  s) to km/(h  s)

mi/(h  s) to knots/s

knots/s to mi/(h  s)

knots/s to km/(h  s)

1 2 3 4 5

118.1 236.2 354.3 472.4 590.6

0.6214 1.243 1.864 2.485 3.107

0.5400 1.080 1.620 2.160 2.700

0.6818 1.364 2.045 2.727 3.409

0.4572 0.9144 1.372 1.829 2.286

0.008467 0.01693 0.02540 0.03387 0.04233

1.609 3.219 4.828 6.437 8.046

0.8690 1.738 2.607 3.476 4.345

1.151 2.302 3.452 4.603 5.754

1.852 3.704 5.556 7.408 9.260

6 7 8 9

708.7 826.8 944.9 1063

3.728 4.350 4.971 5.592

3.240 3.780 4.320 4.860

4.091 4.772 5.454 6.136

2.743 3.200 3.658 4.115

0.05080 0.05927 0.06774 0.07620

9.656 11.27 12.87 14.48

5.214 6.083 6.952 7.821

6.905 8.056 9.206 10.36

11.11 12.96 14.82 16.67

* EXAMPLE: 1 cm/s2  118.1 ft/min2.

Table 1.2.26

Energy or Work Equivalents

Joules or Newton-metres

Kilogramfmetres

1 9.80665 1.356 3.600  106 2.648  106 2.6845  106 101.33 4186.8 1055

0.10197 1 0.1383 3.671  105 270000 2.7375  105 10.333 426.9 107.6

Table 1.2.27

Foot-poundsf

Kilowatt hours

Metric horsepowerhours

Horsepowerhours

Litreatmospheres

Kilocalories

British thermal units

0.7376 7.233 1 2.655  106 1.9529  106 1.98  106 74.74 3088 778.2

0.062778 0.052724 0.063766 1 0.7355 0.7457 0.042815 0.001163 0.032931

0.063777 0.0537037 0.0651206 1.3596 1 1.0139 0.043827 0.001581 0.033985

0.063725 0.053653 0.0650505 1.341 0.9863 1 0.043775 0.001560 0.033930

0.009869 0.09678 0.01338 35528 26131 26493 1 41.32 10.41

0.032388 0.002342 0.033238 859.9 632.4 641.2 0.02420 1 0.25200

0.039478 0.009295 0.001285 3412 2510 2544 0.09604 3.968 1

Conversion of Energy, Work, Heat*

Ft  lbf to joules

Joules to ft  lbf

Ft  lbf to Btu

Btu to ft  lbf

Kilogramfmetres to kilocalories

Kilocalories to kilogramfmetres

Joules to calories

Calories to joules

1 2 3 4

1.3558 2.7116 4.0674 5.4232

0.7376 1.4751 2.2127 2.9503

0.001285 0.002570 0.003855 0.005140

778.2 1,556 2,334 3,113

0.002342 0.004685 0.007027 0.009369

426.9 853.9 1,281 1,708

0.2388 0.4777 0.7165 0.9554

4.187 8.374 12.56 16.75

5 6 7 8 9

6.7790 8.1348 9.4906 10.8464 12.2022

3.6879 4.4254 5.1630 5.9006 6.6381

0.006425 0.007710 0.008995 0.01028 0.01156

3,891 4,669 5,447 6,225 7,003

0.01172 0.01405 0.01640 0.01874 0.02108

2,135 2,562 2,989 3,415 3,842

1.194 1.433 1.672 1.911 2.150

20.93 25.12 29.31 33.49 37.68

* EXAMPLE: 1 ft  lbf  1.3558 J.

Table 1.2.28

Power Equivalents

Horsepower

Kilowatts

Metric horsepower

Kgf  m per s

Ft  lbf per s

Kilocalories per s

Btu per s

1 1.341 0.9863 0.01315 0.00182 5.615 1.415

0.7457 1 0.7355 0.009807 0.001356 4.187 1.055

1.014 1.360 1 0.01333 0.00184 5.692 1.434

76.04 102.0 75 1 0.1383 426.9 107.6

550 737.6 542.5 7.233 1 3088 778.2

0.1781 0.2388 0.1757 0.002342 0.033238 1 0.2520

0.7068 0.9478 0.6971 0.009295 0.001285 3.968 1

1-34

MEASURING UNITS Table 1.2.29

Conversion of Power*

Horsepower to kilowatts

Kilowatts to horsepower

Metric horsepower to kilowatts

Kilowatts to metric horsepower

Horsepower to metric horsepower

Metric horsepower to horsepower

1 2 3 4

0.7457 1.491 2.237 2.983

1.341 2.682 4.023 5.364

0.7355 1.471 2.206 2.942

1.360 2.719 4.079 5.438

1.014 2.028 3.042 4.055

0.9863 1.973 2.959 3.945

5 6 7 8 9

3.729 4.474 5.220 5.966 6.711

6.705 8.046 9.387 10.73 12.07

3.677 4.412 5.147 5.883 6.618

6.798 8.158 9.520 10.88 12.24

5.069 6.083 7.097 8.111 9.125

4.932 5.918 6.904 7.891 8.877

* EXAMPLE: 1 hp  0.7457 kW.

Table 1.2.30

Table 1.2.31

Density Equivalents*

Grams per mL

Lb per cu in

Lb per cu ft

Short tons (2,000 lb) per cu yd

1 27.68 0.01602 1.187 0.1198

0.03613 1 0.035787 0.04287 0.004329

62.43 1728 1 74.7 7.481

0.8428 23.33 0.0135 1 0.1010

Lb per U.S. gal 8.345 231 0.1337 9.902 1

* EXAMPLE: 1 g per mL  62.43 lb per cu ft.

Table 1.2.32

Conversion of Density Grams per mL to lb per cu ft

Lb per cu ft to grams per mL

Grams per mL to short tons per cu yd

Short tons per cu yd to grams per mL

1 2

62.43 124.86

0.01602 0.03204

0.8428 1.6856

1.187 2.373

3 4

187.28 249.71

0.04805 0.06407

2.5283 3.3711

3.560 4.746

5 6

312.14 374.57

0.08009 0.09611

4.2139 5.0567

5.933 7.119

7 8

437.00 499.43

0.11213 0.12814

5.8995 6.7423

8.306 9.492

9 10

561.85 624.28

0.14416 0.16018

7.5850 8.4278

10.679 11.866

Thermal Conductivity

Calories per cm  s  8C

Watts per cm  8C

Calories per cm  h  8C

Btu  ft per ft2  h  8F

Btu  in per ft2  day  8F

1 0.2388 0.0002778 0.004134 0.00001435

4.1868 1 0.001163 0.01731 0.00006009

3,600 860 1 14.88 0.05167

241.9 57.79 0.0672 1 0.00347

69,670 16,641 19.35 288 1

Table 1.2.33

Thermal Conductance

Calories per cm2  s  8C

Watts per cm2  8C

Calories per cm2  h  8C

Btu per ft2  h  8F

Btu per ft2  day  8F

1 0.2388 0.0002778 0.0001356 0.000005651

4.1868 1 0.001163 0.0005678 0.00002366

3,600 860 1 0.4882 0.02034

7,373 1,761 2.048 1 0.04167

176,962 42,267 49.16 24 1

Table 1.2.34

Heat Flow

Calories per cm2  s

Watts per cm2

Calories per cm2  h

Btu per ft2  h

Btu per ft2  day

1 0.2388 0.0002778 0.00007535 0.000003139

4.1868 1 0.001163 0.0003154 0.00001314

3,600 860 1 0.2712 0.01130

13,272 3,170 3.687 1 0.04167

318,531 76,081 88.48 24 1

Section

2

Mathematics BY

C. EDWARD SANDIFER Professor, Western Connecticut State University, Danbury, CT. THOMAS J. COCKERILL Advisory Engineer, International Business Machines, Inc.

2.1 MATHEMATICS by C. Edward Sandifer Sets, Numbers, and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Significant Figures and Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Geometry, Areas, and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Analytical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18 Differential and Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-24 Series and Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-30 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34 Theorems about Line and Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . 2-35

Laplace and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38 2.2 COMPUTERS by Thomas J. Cockerill Computer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 Computer Data Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-41 Computer Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43 Distributed Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45 Relational Database Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Software Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-48 Software Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-50

2-1

2.1 MATHEMATICS by C. Edward Sandifer REFERENCES: Conte and DeBoor, “Elementary Numerical Analysis: An Algorithmic Approach,” McGraw-Hill. Boyce and DiPrima, “Elementary Differential Equations and Boundary Value Problems,” Wiley. Hamming, “Numerical Methods for Scientists and Engineers,” McGraw-Hill. Kreyszig, “Advanced Engineering Mathematics,” Wiley.

The union has the properties: A#A´B

A ¨ B 5 5x : x  A and x  B6

Sets and Elements

The concept of a set appears throughout modern mathematics. A set is a well-defined list or collection of objects and is generally denoted by capital letters, A, B, C, . . . . The objects composing the set are called elements and are denoted by lowercase letters, a, b, x, y, . . . . The notation xA is read “x is an element of A” and means that x is one of the objects composing the set A. There are two basic ways to describe a set. The first way is to list the elements of the set. A 5 52, 4, 6, 8, 106 This often is not practical for very large sets. The second way is to describe properties which determine the elements of the set. A 5 5even numbers from 2 to 106 This method is sometimes awkward since a single set may sometimes be described in several different ways. In describing sets, the symbol : is read “such that.” The expression B 5 5x : x is an even integer, x . 1, x , 116 is read “B equals the set of all x such that x is an even integer, x is greater than 1, and x is less than 11.” Two sets, A and B, are equal, written A  B, if they contain exactly the same elements. The sets A and B above are equal. If two sets, X and Y, are not equal, it is written X  Y. Subsets A set C is a subset of a set A, written C # A, if each element in C is also an element in A. It is also said that C is contained in A. Any set is a subset of itself. That is, A # A always. A is said to be an “improper subset of itself.” Otherwise, if C # A and C  A, then C is a proper subset of A. Two theorems are important about subsets: (Fundamental theorem of set equality) and

Y # X,

then X 5 Y

(2.1.1)

(Transitivity) If X # Y

and

Y # Z,

then X # Z

(2.1.2)

Universe and Empty Set In an application of set theory, it often happens that all sets being considered are subsets of some fixed set, say integers or vectors. This fixed set is called the universe and is sometimes denoted U. It is possible that a set contains no elements at all. The set with no elements is called the empty set or the null set and is denoted [. Set Operations New sets may be built from given sets in several ways. The union of two sets, denoted A ´ B, is the set of all elements belonging to A or to B, or to both.

A ´ B 5 5x : x  A or x  B6

2-2

(2.1.3)

The intersection is denoted A ¨ B and consists of all elements, each of which belongs to both A and B.

SETS, NUMBERS, AND ARITHMETIC

If X # Y

B#A´B

and

The intersection has the properties A¨B#A

A¨B#B

and

(2.1.4)

If A ¨ B 5 [, then A and B are called disjoint. In general, a union makes a larger set and an intersection makes a smaller set. The complement of a set A is the set of all elements in the universe set which are not in A. This is written A 5 5x : x  U, x  A6

,

The difference of two sets, denoted A  B, is the set of all elements which belong to A but do not belong to B. Algebra on Sets The operations of union, intersection, and complement obey certain laws known as Boolean algebra. Using these laws, it is possible to convert an expression involving sets into other equivalent expressions. The laws of Boolean algebra are given in Table 2.1.1. Venn Diagrams To give a pictorial representation of a set, Venn diagrams are often used. Regions in the plane are used to correspond to sets, and areas are shaded to indicate unions, intersections, and complements. Examples of Venn diagrams are given in Fig. 2.1.1. The laws of Boolean algebra and their relation to Venn diagrams are particularly important in programming and in the logic of computer searches. Numbers

Numbers are the basic instruments of computation. It is by operations on numbers that calculations are made. There are several different kinds of numbers. Natural numbers, or counting numbers, denoted N, are the whole numbers greater than zero. Sometimes zero is included as a natural number. Any two natural numbers may be added or multiplied to give another natural number, but subtracting them may produce a negative Table 2.1.1

Laws of Boolean Algebra

1. Idempotency A´A5A 2. Associativity sA ´ Bd ´ C 5 A ´ sB ´ Cd 3. Commutativity A´B5B´A 4. Distributivity A ´ sB ¨ Cd 5 sA ´ Bd ¨ sA ´ Cd 5. Identity A´[5A A´U5U 6. Complement A ´ ,A 5 U , , s Ad 5 A , U5[ , [5U 7. DeMorgan’s laws , sA ´ Bd 5 ,A ¨ ,B

A¨A5A sA ¨ Bd ¨ C 5 A ¨ sB ¨ Cd A¨B5B¨A A ¨ sB ´ Cd 5 sA ¨ Bd ´ sA ¨ Cd A¨U5A A¨[5[ A ¨ ,A 5 [

,

sA ¨ Bd 5 ,A ´ ,B

SETS, NUMBERS, AND ARITHMETIC

2-3

If two functions f and g have the same range and domain and if the ranges are numbers, then f and g may be added, subtracted, multiplied, or divided according to the rules of the range. If f(x)  3x  4 and g(x)  sin(x) and both have range and domain equal to R, then f 1 gsxd 5 3x 1 4 1 sin sxd f 3x 1 4 g sxd 5 sin x Dividing functions occasionally leads to complications when one of the functions assumes a value of zero. In the example f/g above, this occurs when x  0. The quotient cannot be evaluated for x  0 although the quotient function is still meaningful. In this case, the function f/g is said to have a pole at x  0. Polynomial functions are functions of the form and

n

f sxd 5 g ai x i i50

Fig. 2.1.1 Venn diagrams.

number, which is not a natural number, and dividing them may produce a fraction, which is not a natural number. When computers are used, it is important to know what kind of number is being used so the correct data type will be used as well. Integers, or whole numbers, are denoted by Z. They include both positive and negative numbers and zero. Integers may be added, subtracted, and multiplied, but division might not produce an integer. Real numbers, denoted R, are essentially all values which it is possible for a measurement to take, or all possible lengths for line segments. Rational numbers are real numbers that are the quotient of two integers, for example, 11⁄78. Irrational numbers are not the quotient of two integers, for example, p and 22. Within the real numbers, it is always possible to add, subtract, multiply, and divide (except division by zero). Complex numbers, or imaginary numbers, denoted C, are an extension of the real numbers that include the square root of 1, denoted i. Within the real numbers, only positive numbers have square roots. Within the complex numbers, all numbers have square roots. Any complex number z can be written uniquely as z  x  iy, where x and y are real. Then x is the real part of z, denoted Re(z), and y is the imaginary part, denoted Im(z). The complex conjugate, or simply conjugate of a complex number, z is z 5 x 2 iy. If z  x  iy and w  u  iv, then z and w may be manipulated as follows: z 1 w 5 sx 1 ud 1 is y 1 vd z 2 w 5 sx 2 ud 1 is y 2 vd zw 5 xu 2 yv 1 isxv 1 yud xu 1 yv 1 is yu 2 xvd z w5 u2 1 v2 As sets, the following relation exists among these different kinds of numbers: N#Z#R#C Functions A function f is a rule that relates two sets A and B. Given an element x

of the set A, the function assigns a unique element y from the set B. This is written y 5 fsxd The set A is called the domain of the function, and the set B is called the range. It is possible for A and B to be the same set. Functions are usually described by giving the rule. For example, f sxd 5 3x 1 4 is a rule for a function with range and domain both equal to R. Given a value, say, 2, from the domain, f (2)  3(2)  4  10.

where an  0. The domain and range of polynomial functions are always either R or C. The number n is the degree of the polynomial. Polynomials of degree 0 or 1 are called linear; of degree 2 they are called parabolic or quadratic; and of degree 3 they are called cubic. The values of f for which f(x)  0 are called the roots of f. A polynomial of degree n has at most n roots. There is exactly one exception to this rule: If f(x)  0 is the constant zero function, the degree of f is zero, but f has infinitely many roots. Roots of polynomials of degree 1 are found as follows: Suppose the polynomial is f(x)  ax  b. Set f(x)  0 and solve for x. Then x  b/a. Roots of polynomials of degree 2 are often found using the quadratic formula. If f(x)  a x2  bx  c, then the two roots of f are given by the quadratic formula:

2b 1 2b 2 2 4ac 2b 2 2b 2 2 4ac and x2 5 2a 2a Roots of a polynomial of degree 3 fall into two types. x1 5

Equations of the Third Degree with Term in x 2 Absent

Solution: After dividing through by the coefficient of x3, any equation of this type can be written x3  Ax  B. Let p  A/3 and q  B/2. The general solution is as follows: CASE 1. q2  p3 positive. One root is real, viz., 3 x1 5 #q 1 2q 2 2 p 3 1 # q 2 2q 2 2 p 3 3

The other two roots are imaginary. CASE 2.

q2  p3  zero. Three roots real, but two of them equal. 3 x1 5 2 2 q

3 x2 5 2 2 q

3 x3 5 2 2 q

CASE 3. q2  p3 negative. All three roots are real and distinct. Determine an angle u between 0 and 180, such that cos u 5 q/s p 2pd. Then

x1 5 2 2p cos su/3d x2 5 2 2p cos su/3 1 1208d x3 5 2 2p cos su/3 1 2408d Graphical Solution: Plot the curve y1  x3, and the straight line y2  Ax  B. The abscissas of the points of intersection will be the roots of the equation. Equations of the Third Degree (General Case)

Solution: The general cubic equation, after dividing through by the coefficient of the highest power, may be written x3  ax2  bx  c  0. To get rid of the term in x2, let x  x1  a/3. The equation then becomes x 31 5 Ax1 1 B, where A  3(a/3)2  b, and B  2(a/3)3  b(a/3)  c. Solve this equation for x1, by the method above, and then find x itself from x  x1  (a/3). Graphical Solution: Without getting rid of the term in x2, write the equation in the form x3  a[x  (b/ 2a)]2  [a(b/ 2a)2  c], and solve by the graphical method.

2-4

MATHEMATICS

Computer Solutions: Equations of degree 3 or higher are frequently solved by using computer algebra systems such as Mathematica or Maple.

0.6  0.8  .048 should be written as 0.5 since the factors have one significant figure. There is a gain of precision from 0.1 to 0.01.

Arithmetic

Addition and Subtraction A sum or difference should be represented with the same precision as the least precise term involved. The number of significant figures may change.

When numbers, functions, or vectors are manipulated, they always obey certain properties, regardless of the types of the objects involved. Elements may be added or subtracted only if they are in the same universe set. Elements in different universes may sometimes be multiplied or divided, but the result may be in a different universe. Regardless of the universe sets involved, the following properties hold true: 1. Associative laws. a  (b  c)  (a  b)  c, a(bc)  (ab)c 2. Identity laws. 0  a  a, 1a  a 3. Inverse laws. a  a  0, a/a  1 4. Distributive law. a(b  c)  ab  ac 5. Commutative laws. a  b  b  a, ab  ba Certain universes, for example, matrices, do not obey the commutative law for multiplication. SIGNIFICANT FIGURES AND PRECISION Number of Significant Figures In engineering computations, the data are ordinarily the result of measurement and are correct only to a limited number of significant figures. Each of the numbers 3.840 and 0.003840 is said to be given “correct to four figures”; the true value lies in the first case between 0.0038395 and 0.0038405. The absolute error is less than 0.001 in the first case, and less than 0.000001 in the second; but the relative error is the same in both cases, namely, an error of less than “one part in 3,840.” If a number is written as 384,000, the reader is left in doubt whether the number of correct significant figures is 3, 4, 5, or 6. This doubt can be removed by writing the number as 3.84  10 5, or 3.840  10 5, or 3.8400  10 5, or 3.84000  10 5. In any numerical computation, the possible or desirable degree of accuracy should be decided on and the computation should then be so arranged that the required number of significant figures, and no more, is secured. Carrying out the work to a larger number of places than is justified by the data is to be avoided, (1) because the form of the results leads to an erroneous impression of their accuracy and (2) because time and labor are wasted in superfluous computation. The unit value of the least significant figure in a number is its precision. The number 123.456 has six significant figures and has precision 0.001. Two ways to represent a real number are as fixed-point or as floatingpoint, also known as “scientific notation.” In scientific notation, a number is represented as a product of a mantissa and a power of 10. The mantissa has its first significant figure either immediately before or immediately after the decimal point, depending on which convention is being used. The power of 10 used is called the exponent. The number 123.456 may be represented as either

0.123456 3 103

or

1.23456 3 102

Fixed-point representations tend to be more convenient when the quantities involved will be added or subtracted or when all measurements are taken to the same precision. Floating-point representations are more convenient for very large or very small numbers or when the quantities involved will be multiplied or divided. Many different numbers may share the same representation. For example, 0.05 may be used to represent, with precision 0.01, any value between 0.045000 and 0.054999. The largest value a number represents, in this case 0.0549999, is sometimes denoted x*, and the smallest is denoted x*. An awareness of precision and significant figures is necessary so that answers correctly represent their accuracy. Multiplication and Division A product or quotient should be written with the smallest number of significant figures of any of the factors involved. The product often has a different precision than the factors, but the significant figures must not increase. EXAMPLES. (6.)(8.)  48 should be written as 50 since the factors have one significant figure. There is a loss of precision from 1 to 10.

EXAMPLES. 3.14  0.001  3.141 should be represented as 3.14 since the least precise term has precision 0.01. 3.14  0.1  3.24 should be represented as 3.2 since the least precise term has precision 0.1. Loss of Significant Figures Addition and subtraction may result in serious loss of significant figures and resultant large relative errors if the sums are near zero. For example,

3.15  3.14  0.01 shows a loss from three significant figures to just one. Where it is possible, calculations and measurements should be planned so that loss of significant figures can be avoided. Mixed Calculations When an expression involves both products and sums, significant figures and precision should be noted for each term or factor as it is calculated, so that correct significant figures and precision for the result are known. The calculation should be performed to as much precision as is available and should be rounded to the correct precision when the calculation is finished. This process is frequently done incorrectly, particularly when calculators or computers provide many decimal places in their result but provide no clue as to how many of those figures are significant. Significant Figures in Evaluating Functions If y  f(x), then the correct number of significant figures in y depends on the number of significant figures in x and on the behavior of the function f in the neighborhood of x. In general, y should be represented so that all of f(x), f(x*), and f (x*) are between y* and y*. EXAMPLES. 5 5 5 5

1.39642 1.41421 1.43178 1.4

sin s0.5d sin s1.0d sin s1.5d so sin s18d

5 5 5 5

0.00872 0.01745 0.02617 0.0

sin s89.5d sin s90.0d sin s90.5d so sin s908d

5 5 5 5

0.99996 1.00000 0.99996 1.0000

sqr s2.0d sqr s1.95d sqr s2.00d sqr s2.05d so y sin s18d

sin s908d

Note that in finding sin (90), there was a gain in significant figures from two to five and also a gain in precision. This tends to happen when f(x) is close to zero. On the other hand, precision and significant figures are often lost when f (x) or f (x) are large. Rearrangement of Formulas Often a formula may be rewritten in order to avoid a loss of significant figures. In using the quadratic formula to find the roots of a polynomial, significant figures may be lost if the ax2  bx  c has a root near zero. The quadratic formula may be rearranged as follows: 1. Use the quadratic formula to find the root that is not close to 0. Call this root x1. 2. Then x2  c/ax1. If f sxd 5 2x 1 1 2 2x, then loss of significant figures occurs if x is large. This can be eliminated by “rationalizing the numerator” as follows: s 2x 1 1 2 2xds 2x 1 1 1 2xd 2x 1 1 1 2x and this has no loss of significant figures.

5

1 2x 1 1 1 2x

GEOMETRY, AREAS, AND VOLUMES

There is an almost unlimited number of “tricks” for rearranging formulas to avoid loss of significant figures, but many of these are very similar to the tricks used in calculus to evaluate limits.

2-5

The Circle An angle that is inscribed in a semicircle is a right angle (Fig. 2.1.6). A tangent is perpendicular to the radius drawn to the point of contact.

GEOMETRY, AREAS, AND VOLUMES Geometrical Theorems Right Triangles a2  b2  c2. (See Fig. 2.1.2.) / A 1 / B 5 908. p2  mn. a2  mc. b2  nc. Oblique Triangles Sum of angles  180. An exterior angle  sum of the two opposite interior angles (Fig. 2.1.2).

Fig. 2.1.2 Right triangle.

The medians, joining each vertex with the middle point of the opposite side, meet in the center of gravity G (Fig. 2.1.3), which trisects each median.

Fig. 2.1.6 Angle inscribed in a semicircle.

Fig. 2.1.7 Dihedral angle.

Dihedral Angles The dihedral angle between two planes is measured by a plane angle formed by two lines, one in each plane, perpendicular to the edge (Fig. 2.1.7). (For solid angles, see Surfaces and Volumes of Solids.) In a tetrahedron, or triangular pyramid, the four medians, joining each vertex with the center of gravity of the opposite face, meet in a point, the center of gravity of the tetrahedron; this point is 3⁄4 of the way from any vertex to the center of gravity of the opposite face. The Sphere (See also Surfaces and Volumes of Solids.) If AB is a diameter, any plane perpendicular to AB cuts the sphere in a circle, of which A and B are called the poles. A great circle on the sphere is formed by a plane passing through the center. Geometrical Constructions To Bisect a Line AB (Fig. 2.1.8) (1) From A and B as centers, and with equal radii, describe arcs intersecting at P and Q, and draw PQ, which will bisect AB in M. (2) Lay off AC  BD  approximately half of AB, and then bisect CD.

Fig. 2.1.3 Triangle showing medians and center of gravity.

The altitudes meet in a point called the orthocenter, O. The perpendiculars erected at the midpoints of the sides meet in a point C, the center of the circumscribed circle. (In any triangle G, O, and C lie in line, and G is two-thirds of the way from O to C.) The bisectors of the angles meet in the center of the inscribed circle (Fig. 2.1.4).

Fig. 2.1.4 Triangle showing bisectors of angles.

The largest side of a triangle is opposite the largest angle; it is less than the sum of the other two sides. Similar Figures Any two similar figures, in a plane or in space, can be placed in “perspective,” i.e., so that straight lines joining corresponding points of the two figures will pass through a common point (Fig. 2.1.5). That is, of two similar figures, one is merely an enlargement of the other. Assume that each length in one figure is k times the corresponding length in the other; then each area in the first figure is k2 times the corresponding area in the second, and each volume in the first figure is k3 times the corresponding volume in the second. If two lines are cut by a set of parallel lines (or parallel planes), the corresponding segments are proportional.

Fig. 2.1.8 Bisectors of a line.

Fig. 2.1.9 Construction of a line parallel to a given line.

To Draw a Parallel to a Given Line l through a Given Point A (Fig. 2.1.9) With point A as center draw an arc just touching the line l; with any point O of the line as center, draw an arc BC with the same radius. Then a line through A touching this arc will be the required parallel. Or, use a straightedge and triangle. Or, use a sheet of celluloid with a set of lines parallel to one edge and about 1⁄4 in apart ruled upon it. To Draw a Perpendicular to a Given Line from a Given Point A Outside the Line (Fig. 2.1.10) (1) With A as center, describe an arc

cutting the line at R and S, and bisect RS at M. Then M is the foot of the perpendicular. (2) If A is nearly opposite one end of the line, take any point B of the line and bisect AB in O; then with O as center, and OA or OB as radius, draw an arc cutting the line in M. Or, (3) use a straightedge and triangle.

Fig. 2.1.10 Construction of a line perpendicular to a given line from a point not on the line.

Fig. 2.1.5 Similar figures.

To Erect a Perpendicular to a Given Line at a Given Point P (1) Lay off PR  PS (Fig. 2.1.11), and with R and S as centers draw arcs

2-6

MATHEMATICS

intersecting at A. Then PA is the required perpendicular. (2) If P is near the end of the line, take any convenient point O (Fig. 2.1.12) above the line as center, and with radius OP draw an arc cutting the line at Q. Produce QO to meet the arc at A; then PA is the required perpendicular. (3) Lay off PB  4 units of any scale (Fig. 2.1.13); from P and B as centers lay off PA  3 and BA  5; then APB is a right angle.

Fig. 2.1.11 Construction of a line perpendicular to a given line from a point on the line.

Fig. 2.1.12 Construction of a line perpendicular to a given line from a point on the line.

To Divide a Line AB into n Equal Parts (Fig. 2.1.14) Through A draw a line AX at any angle, and lay off n equal steps along this line. Connect the last of these divisions with B, and draw parallels through the other divisions. These parallels will divide the given line into n equal parts. A similar method may be used to divide a line into parts which shall be proportional to any given numbers. To Bisect an Angle AOB (Fig. 2.1.15) Lay off OA  OB. From A and B as centers, with any convenient radius, draw arcs meeting at M; then OM is the required bisector.

Fig. 2.1.13 Construction of a line perpendicular to a given line from a point on the line.

Fig. 2.1.14 Division of a line into equal parts.

To draw the bisector of an angle when the vertex of the angle is not accessible. Parallel to the given lines a, b, and equidistant from them, draw two lines a, b which intersect; then bisect the angle between a and b. To Inscribe a Hexagon in a Circle (Fig. 2.1.16) Step around the circumference with a chord equal to the radius. Or, use a 60 triangle.

Fig. 2.1.17 Hexagon circumscribed about a circle.

Fig. 2.1.18 Construction of a polygon with a given side.

To Draw a Common Tangent to Two Given Circles (Fig. 2.1.20) Let C and c be centers and R and r the radii (R  r). From C as center, draw two concentric circles with radii R  r and R  r; draw tangents to

Fig. 2.1.19 Construction of a tangent to a circle.

Fig. 2.1.20 Construction of a tangent common to two circles.

these circles from c; then draw parallels to these lines at distance r. These parallels will be the required common tangents. To Draw a Circle through Three Given Points A, B, C, or to find the center of a given circular arc (Fig. 2.1.21) Draw the perpendicular bisectors of AB and BC; these will meet at the center, O.

Fig. 2.1.21 Construction of a circle passing through three given points. To Draw a Circle through Two Given Points A, B, and Touching a Given Circle (Fig. 2.1.22) Draw any circle through A and B, cutting

the given circle at C and D. Let AB and CD meet at E, and let ET be tangent from E to the circle just drawn. With E as center, and radius ET, draw an arc cutting the given circle at P and Q. Either P or Q is the required point of contact. (Two solutions.) Fig. 2.1.15 Bisection of an angle.

Fig. 2.1.16 Hexagon inscribed in a circle.

To Circumscribe a Hexagon about a Circle (Fig. 2.1.17) Draw a chord AB equal to the radius. Bisect the arc AB at T. Draw the tangent at T (parallel to AB), meeting OA and OB at P and Q. Then draw a circle with radius OP or OQ and inscribe in it a hexagon, one side being PQ. To Construct a Polygon of n Sides, One Side AB Being Given

(Fig. 2.1.18) With A as center and AB as radius, draw a semicircle, and divide it into n parts, of which n  2 parts (counting from B) are to be used. Draw rays from A through these points of division, and complete the construction as in the figure (in which n  7). Note that the center of the polygon must lie in the perpendicular bisector of each side. To Draw a Tangent to a Circle from an external point A (Fig. 2.1.19) Bisect AC in M; with M as center and radius MC, draw arc cutting circle in P; then P is the required point of tangency.

Fig. 2.1.22 Construction of a circle through two given points and touching a given circle. To Draw a Circle through One Given Point, A, and Touching Two Given Circles (Fig. 2.1.23) Let S be a center of similitude for the two

given circles, i.e., the point of intersection of two external (or internal)

GEOMETRY, AREAS, AND VOLUMES

common tangents. Through S draw any line cutting one circle at two points, the nearer of which shall be called P, and the other at two points, the more remote of which shall be called Q. Through A, P, Q draw a circle cutting SA at B. Then draw a circle through A and B and touching one of the given circles (see preceding construction). This circle will touch the other given circle also. (Four solutions.)

2-7

Rectangle (Fig. 2.1.28) Area  ab  1⁄2D2 sin u, where u  angle between diagonals D, D. Rhombus (Fig. 2.1.29) Area  a2 sin C  1⁄2D1D2, where C  angle between two adjacent sides; D1, D2  diagonals.

Fig. 2.1.28 Rectangle.

Fig. 2.1.29 Rhombus.

Parallelogram (Fig. 2.1.30) Area  bh  ab sin C  1⁄2D1D2 sin u, where u  angle between diagonals D1 and D2. Trapezoid (Fig. 2.1.31) Area  1⁄2(a  b) h where bases a and b are parallel. Fig. 2.1.23 Construction of a circle through a given point and touching two given circles. To Draw an Annulus Which Shall Contain a Given Number of Equal Contiguous Circles (Fig. 2.1.24) (An annulus is a ring-shaped area

enclosed between two concentric circles.) Let R  r and R  r be the inner and outer radii of the annulus, r being the radius of each of the n circles. Then the required relation between these quantities is given by r  R sin (180/n), or r  (R  r) [sin (180/n)]/[1  sin (180/n)].

Fig. 2.1.30 Parallelogram.

Fig. 2.1.31 Trapezoid.

Any Quadrilateral (Fig. 2.1.32) Area  1⁄2D1D2 sin u.

Fig. 2.1.24 Construction of an annulus containing a given number of contiguous circles. Fig. 2.1.32 Quadrilateral. Lengths and Areas of Plane Figures Right Triangle (Fig. 2.1.25) a2  b2  c2. Area  1⁄2ab  1⁄2a2 cot A  1⁄2 b2 tan A  1⁄4c2 sin 2A. Equilateral Triangle (Fig. 2.1.26) Area 5 1⁄4 a 2 23 5 0.43301a 2.

Fig. 2.1.25 Right triangle.

Fig. 2.1.26 Equilateral triangle.

Regular Polygons n  number of sides; v  360/n  angle subtended at center by one side; a  length of one side  2R sin (v/2)  2r tan (v/2); R  radius of circumscribed circle  0.5 a csc (v/2)  r sec (v/2); r  radius of inscribed circle  R cos (v/2)  0.5 cot (v/2); area  0.25 a2n cot (v/2)  0.5 R2n sin (v)  r2n tan (v/2). Areas of regular polygons are tabulated in Table 1.1.3. Circle Area  pr2  1⁄2Cr  1⁄4Cd  1⁄4pd 2  0.785398d 2, where r  radius, d  diameter, C  circumference  2 pr  pd. Annulus (Fig. 2.1.33) Area  p(R2  r 2 )  p(D2  d 2 )/4  2pRb, where R  mean radius  1⁄2(R  r), and b  R  r.

Any Triangle (Fig. 2.1.27)

s 5 1⁄2 sa 1 b 1 cd, t 5 1⁄2 sm1 1 m2 1 m3d r 5 2ss 2 adss 2 bdss 2 cd/s 5 radius inscribed circle R 5 1⁄2 a/sin A 5 1⁄2 b/sin B 5 1⁄2 c/sin C 5 radius circumscribed circle Area 5 1⁄2 base 3 altitude 5 1⁄2 ah 5 1⁄2 ab sin C 5 rs 5 abc/4R 5 61⁄2 5 sx1 y2 2 x2 y1d 1 sx2 y3 2 x3 y2d 1 sx3 y1 2 x1 y3d6, where sx1, y1d, sx2, y2d, sx3, y3d are coordinates of vertices.

Fig. 2.1.33 Annulus.

(Fig. 2.1.34) Area  1⁄2rs  pr2A/360  1⁄2r2 rad A, where rad A  radian measure of angle A, and s  length of arc  r rad A. Sector

Fig. 2.1.27 Triangle.

Fig. 2.1.34 Sector.

2-8

MATHEMATICS

Segment (Fig. 2.1.35) Area  1⁄2r2(rad A  sin A)  1⁄2[r(s  c)  ch], where rad A radian measure of angle A. For small arcs, s  1⁄3(8c  c), where c  chord of half of the arc (Huygens’ approximation). Areas of segments are tabulated in Tables 1.1.1 and 1.1.2.

Right Circular Cylinder (Fig. 2.1.40) Volume  pr2h  Bh. Lateral area  2prh  Ph. Here B  area of base; P  perimeter of base.

Fig. 2.1.39 Regular prism.

Fig. 2.1.35 Segment. Ribbon bounded by two parallel curves (Fig. 2.1.36). If a straight line AB moves so that it is always perpendicular to the path traced by its middle point G, then the area of the ribbon or strip thus generated is equal to the length of AB times the length of the path traced by G. (It is assumed that the radius of curvature of G’s path is never less than 1⁄2AB, so that successive positions of generating line will not intersect.)

Fig. 2.1.40 Right circular cylinder.

Truncated Right Circular Cylinder (Fig. 2.1.41) Volume  pr2h  Bh. Lateral area  2prh  Ph. Here h  mean height  1⁄2(h  h ); B  area of base; P  perimeter of base. 1 2

Fig. 2.1.41 Truncated right circular cylinder. Fig. 2.1.36 Ribbon. Ellipse (Fig. 2.1.37) Area of ellipse  pab. Area of shaded segment  xy  ab sin1 (x/a). Length of perimeter of ellipse  p(a  b)K, where K  (1  1⁄4m2  1⁄64m4  1⁄256m6  . . .), m  (a  b)/(a  b).

For m  0.1 K  1.002 For m  0.6 K  1.092

0.2 1.010 0.7 1.127

0.3 1.023 0.8 1.168

0.4 1.040 0.9 1.216

Any Prism or Cylinder (Fig. 2.1.42) Volume  Bh  Nl. Lateral area  Ql. Here l  length of an element or lateral edge; B  area of base; N  area of normal section; Q  perimeter of normal section.

0.5 1.064 1.0 1.273 Fig. 2.1.42 Any prism or cylinder. Special Ungula of a Right Cylinder (Fig. 2.1.43) Volume  2⁄3r2H. Lateral area  2rH. r  radius. (Upper surface is a semiellipse.)

Fig. 2.1.37 Ellipse. Hyperbola (Fig. 2.1.38) In any hyperbola, shaded area A  ab ln [(x/a)  (y/b)]. In an equilateral hyperbola (a  b), area A  a2 sinh1 (y/a)  a2 cosh1 (x/a). Here x and y are coordinates of point P. Fig. 2.1.43 Special ungula of a right circular cylinder. Any Ungula of a right circular cylinder (Figs. 2.1.44 and 2.1.45) Volume  H(2⁄3 a3 cB)/(r c)  H[a(r2  1⁄3a2) r2c rad u]/ (r c). Lateral area  H(2ra cs)/(r c)  2rH(a c rad u)/

Fig. 2.1.38 Hyperbola.

For lengths and areas of other curves see Analytical Geometry. Surfaces and Volumes of Solids Regular Prism (Fig. 2.1.39) Volume  1⁄2nrah  Bh. Lateral area  nah  Ph. Here n  number of sides; B  area of base; P  perimeter of base.

Fig. 2.1.44 Ungula of a right circular cylinder.

Fig. 2.1.45 Ungula of a right circular cylinder.

GEOMETRY, AREAS, AND VOLUMES

(r c). If base is greater (less) than a semicircle, use  () sign. r  radius of base; B  area of base; s  arc of base; u  half the angle subtended by arc s at center; rad u  radian measure of angle u. Regular Pyramid (Fig. 2.1.46) Volume  1⁄3 altitude  area of base  1⁄6hran. Lateral area  1⁄2 slant height  perimeter of base  1⁄2san. Here r  radius of inscribed circle; a  side (of regular polygon); n  number of sides; s 5 2r 2 1 h2. Vertex of pyramid directly above center of base.

2-9

pd 2  lateral area of circumscribed cylinder. Here r  radius; 3 d 5 2r 5 diameter 5 2 6V/p 5 2A/p. Hollow Sphere or spherical shell. Volume 5 4⁄3psR 3 2 r 3d  1⁄6psD 3 2 d 3d 5 4pR 2 t 1 1⁄3pt 3. Here R, r  outer and inner radii; 1 D,d  outer and inner diameters; t  thickness  R  r; R1  mean radius  1⁄2(R  r). Any Spherical Segment. Zone (Fig. 2.1.50) Volume  1⁄6phs3a 2 1 3a 2 1 h2d. Lateral area (zone)  2prh. Here r  radius of 1 sphere. If the inscribed frustum of a cone is removed from the spherical segment, the volume remaining is 1⁄6phc2, where c  slant height of frustum  2h2 1 sa 2 a1d2.

Fig. 2.1.50 Any spherical segment. Fig. 2.1.46 Regular pyramid. Right Circular Cone Volume  1⁄3pr2h. Lateral area  prs. Here

r  radius of base; h  altitude; s 5 slant height 5 2r 2 1 h2. Frustum of Regular Pyramid (Fig. 2.1.47) Volume  1⁄6hran[1  (a/a)  (a/a)2]. Lateral area  slant height  half sum of perimeters of bases  slant height  perimeter of midsection  1⁄2sn(r  r). Here r,r radii of inscribed circles; s 5 2sr 2 rrd2 1 h2; a,a  sides of lower and upper bases; n  number of sides. Frustum of Right Circular Cone (Fig. 2.1.48) Volume  1⁄3pr 2 h[1  (r/r)  (r/r)2]  1⁄3ph(r 2  rr  r2)  1⁄4ph[r  r)2  1⁄3(r  r)2]. Lateral area 5 p ssr 1 rrd; s 5 2sr 2 rrd2 1 h2.

Spherical Segment of One Base. Zone (spherical “cap” of Fig. 2.1.51) Volume  1⁄6ph(3a2  h2 )  1⁄3ph2 (3r  h). Lateral area (of zone)  2prh  p(a2  h2).

NOTE.

a2  h(2r  h), where r  radius of sphere.

Spherical Sector (Fig. 2.1.51) Volume  1⁄3r  area of cap  ⁄3pr2h. Total area  area of cap  area of cone  2prh  pra.

2

NOTE.

a2  h(2r  h).

Spherical Wedge bounded by two plane semicircles and a lune (Fig. 2.1.52). Volume of wedge volume of sphere  u/360. Area of lune area of sphere  u/360. u  dihedral angle of the wedge.

Fig. 2.1.51 Spherical sector. Fig. 2.1.47 Frustum of a regular pyramid.

Fig. 2.1.52 Spherical wedge.

Fig. 2.1.48 Frustum of a right circular cone.

Any Pyramid or Cone Volume  1⁄3Bh. B  area of base; h 

perpendicular distance from vertex to plane in which base lies. Any Pyramidal or Conical Frustum (Fig. 2.1.49) Volume  1⁄3hsB 1 2BBr 1 Brd 5 1⁄3hB[1 1 sPr/Pd 1 sPr/Pd2]. Here B, B  areas of lower and upper bases; P, P perimeters of lower and upper bases.

Solid Angles Any portion of a spherical surface subtends what is called a solid angle at the center of the sphere. If the area of the given portion of spherical surface is equal to the square of the radius, the subtended solid angle is called a steradian, and this is commonly taken as the unit. The entire solid angle about the center is equal to 4p steradians. A so-called “solid right angle” is the solid angle subtended by a quadrantal (or trirectangular) spherical triangle, and a “spherical degree” (now little used) is a solid angle equal to 1⁄90 of a solid right angle. Hence 720 spherical degrees  1 steregon, or p steradians  180 spherical degrees. If u = the angle which an element of a cone makes with its axis, then the solid angle of the cone contains 2p(1  cos u) steradians. Regular Polyhedra A  area of surface; V  volume; a  edge.

Name of solid Fig. 2.1.49 Pyramidal frustum and conical frustum. Sphere Volume  V  4⁄3p r 3  4.188790r3  1⁄6pd3  2⁄3 volume of circumscribed cylinder. Area  A  4pr2  four great circles 

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Bounded by

A/a2

V/a3

4 triangles 6 squares 8 triangles 12 pentagons 20 triangles

1.7321 6.0000 3.4641 20.6457 8.6603

0.1179 1.0000 0.4714 7.6631 2.1917

2-10

MATHEMATICS

Ellipsoid (Fig. 2.1.53) Volume  4⁄3pabc, where a, b, c  semi-

axes.

Torus, or Anchor Ring (Fig. 2.1.54) Volume  2p2cr2. Area 

4p2cr.

EXAMPLE. The set of four elements (a, b, c, d) has C(4, 2)  6 two-element subsets, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, and {c, d}. (Note that {a, c} is the same set as {c, a}.) Permutations The number of ways k objects may be arranged from a set of n elements is given by

Psn, kd 5

n! sn 2 kd!

EXAMPLE. Two elements from the set (a, b, c, d ) may be arranged in P(4, 2)  12 ways: ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, and dc. Note that ac is a different arrangement than ca. Fig. 2.1.54 Torus.

Fig. 2.1.53 Ellipsoid.

Volume of a Solid of Revolution (solid generated by rotating an area bounded above by f(x) around the x axis) b

V 5 p 3 | f sxd| 2 dx a

Area of a Surface of Revolution b

A 5 2p 3 y 21 1 sdy/dxd2 dx a

Length of Arc of a Plane Curve y  f (x) between values x  a and b

x  b. s 5 3 21 1 sdy/dxd2 dx. If x  f(t) and y  g(t), for a t a b, then

PsA|Ed 5 PsA ¨ Ed/PsEd

b

s 5 3 2sdx/dtd2 1 sdy/dtd2 dt a

PERMUTATIONS AND COMBINATIONS

The product (1)(2)(3) . . . (n) is written n! and is read “n factorial.” By convention, 0!  1, and n! is not defined for negative integers. For large values of n, n! may be approximated by Stirling’s formula: n! < 2.50663nn1.5e2n n The binomial coefficient C(n, k), also written ¢ ≤, is defined as: k Csn, kd

Permutations and combinations are examined in detail in most texts on probability and statistics and on discrete mathematics. If an event can occur in s ways and can fail to occur in f ways, and if all ways are equally likely, then the probability of the event’s occurring is p  s/(s  f ), and the probability of failure is q  f/(s  f )  1  p. The set of all possible outcomes of an experiment is called the sample space, denoted S. Let n be the number of outcomes in the sample set. A subset A of the sample space is called an event. The number of outcomes in A is s. Therefore P(A)  s/n. The probability that A does not occur is P(,A)  q  1  p. Always 0  p  1 and P(S)  1. If two events cannot occur simultaneously, then A ¨ B  , and A and B are said to be mutually exclusive. Then P(A ´ B)  P(A)  P(B). Otherwise, P(A ´ B)  P(A)  P(B)  P(A ¨ B). Events A and B are independent if P(A ¨ B)  P(A)P(B). If E is an event and if P(E)  0, then the probability that A occurs once E has already occurred is called the “conditional probability of A given E,” written P( A|E) and defined as

n! k!sn 2 kd!

C(n, k) is read “n choose k” or as “binomial coefficient n-k.” Binomial coefficients have the following properties: 1. C(n, 0)  C(n, n)  1 2. C(n, 1)  C(n, n  1)  n 3. C(n  1, k)  C(n, k)  C(n, k  1) 4. C(n, k)  C(n, n  k) Binomial coefficients are tabulated in Sec. 1. Binomial Theorem

If n is a positive integer, then n

sa 1 bdn 5 g Csn, kda kb n2k k50

A and E are independent if P( A|E)  P(A). If the outcomes in a sample space X are all numbers, then X, together with the probabilities of the outcomes, is called a random variable. If xi is an outcome, then pi  P(xi). The expected value of a random variable is EsXd 5  ei pi The variance of X is VsXd 5 [xi 2 EsXd]2pi The standard deviation is SsXd 5 2[VsXd] The Binomial, or Bernoulli, Distribution If an experiment is repeated n times and the probability of a success on any trial is p, then the probability of k successes among those n trials is

f sn, k, pd 5 Csn, kdp kq n2k Geometric Distribution If an experiment is repeated until it finally succeeds, let x be the number of failures observed before the first success. Let p be the probability of success on any trial and let q  1  p. Then Psx 5 kd 5 q k # p Uniform Distribution If the random variable x assumes the values 1, 2, . . . , n, with equal probabilities, then the distribution is uniform, and

ExAMPLE. The third term of (2x  3)7 is C(7, 4)(2x)7434  [7!/ (4!3!)](2x)334  (35)(8 x3)(81)  22680x3.

1 Psx 5 kd 5 n

Combinations C(n, k) gives the number of ways k distinct objects can be chosen from a set of n elements. This is the number of k-element subsets of a set of n elements.

Hypergeometric Distribution—Sampling without Replacement If a finite population of N elements contains x successes and if n items are selected randomly without replacement, then the probability that k

LINEAR ALGEBRA

successes will occur among those n samples is Csk, xdCsN 2 k, n 2 xd hsx; N, n, kd 5 CsN, nd For large values of N, the hypergeometric distribution approaches the binomial distribution, so hsx; N, n, kd < f ¢n, k,

x ≤ N

Poisson Distribution If the average number of successes which occur in a given fixed time interval is m, then let x be the number of successes observed in that time interval. The probability that x  k is

psk, md 5

e2mm x x!

Three-dimensional vectors correspond to points in space, where v1, v2, and v3 are the x, y, and z coordinates of the point, respectively. Two- and three-dimensional vectors may be thought of as having a direction and a magnitude. See the section “Analytical Geometry.” Two vectors u and v are equal if: 1. u and v are the same type (either row or column). 2. u and v have the same dimension. 3. Corresponding components are equal; that is, ui  vi for i  1, 2, . . . , n. Note that the row vectors u 5 s1, 2, 3d

b*(k; n, p)  C(k  1, n  1)p q

n kn

The expected values and variances of these distributions are summarized in the following table:

Distribution

E(X)

V(X)

Uniform Binomial Hypergeometric Poisson Geometric Negative binomial

(n  1)/2 np nk/N m q/p nq/p

(n2  1)/12 npq [nk(N  n)(1  k/N)]/[N(N  1)] m q/p2 nq/p2

and

v 5 s3, 2, 1d

are not equal since the components are not in the same order. Also,

where e 5 2.71828 c

Negative Binomial Distribution If repeated independent trials have probability of success p, then let x be the trial number upon which success number n occurs. Then the probability that x  k is

2-11

u 5 s1, 2, 3d

and

1 v 5 £2≥ 3

are not equal since u is a row vector and v is a column vector. Vector Transpose If u is a row vector, then the transpose of u, written uT, is the column vector with the same components in the same order as u. Similarly, the transpose of a column vector is the row vector with the same components in the same order. Note that (uT )T  u. Vector Addition If u and v are vectors of the same type and the same dimension, then the sum of u and v, written u  v, is the vector obtained by adding corresponding components. In the case of row vectors, u 1 v 5 su1 1 v1, u2 1 v2, . . . , un 1 vnd Scalar Multiplication If a is a number and u is a vector, then the scalar product au is the vector obtained by multiplying each component

of u by a. au 5 sau1, au2, . . . , aund A number by which a vector is multiplied is called a scalar. The negative of vector u is written u, and

LINEAR ALGEBRA

Using linear algebra, it is often possible to express in a single equation a set of relations that would otherwise require several equations. Similarly, it is possible to replace many calculations involving several variables with a few calculations involving vectors and matrices. In general, the equations to which the techniques of linear algebra apply must be linear equations; they can involve no polynomial, exponential, or trigonometric terms. Vectors

A row vector v is a list of numbers written in a row, usually enclosed by parentheses. v 5 sv1, v2, c, vnd A column vector u is a list of numbers written in a column: u1 u2 # u5• # µ # un The numbers ui and vi may be real or complex, or they may even be variables or functions. A vector is sometimes called an ordered n-tuple. In the case where n  2, it may be called an ordered pair. The numbers vi are called components or coordinates of the vector v. The number n is called the dimension of v. Two-dimensional vectors correspond with points in the plane, where v1 is the x coordinate and v2 is the y coordinate of the point v. Twodimensional vectors also correspond with complex numbers, where z  v1  iv2.

u  1u The zero vector is the vector with all its components equal to zero. Arithmetic Properties of Vectors If u, v, and w are vectors of the same type and dimensions, and if a and b are scalars, then vector addition and scalar multiplication obey the following seven rules, known as the properties of a vector space: 1. (u  v)  w  u  (v  w) associative law 2. u  v  v  u commutative law 3. u  0  u additive identity 4. u  (u)  0 additive inverse 5. a(u  v)  au  av distributive law 6. (ab)u = a(bu) associative law of multiplication 7. 1u  u multiplicative identity Inner Product or Dot Product If u and v are vectors of the same type and dimension, then their inner product or dot product, written uv or u  v, is the scalar uv 5 u1v1 1 u2v2 1 # # # 1 unvn Vectors u and v are perpendicular or orthogonal if uv  0. Magnitude There are two equivalent ways to define the magnitude of a vector u, written |u| or ||u||. |u| 5 2su ? ud or

|u| 5 2su 21 1 u 22 1 # # # 1 u 2nd

Cross Product or Outer Product If u and v are three-dimensional vectors, then they have a cross product, also called outer product or vector product.

u 3 v 5 su2v3 2 u3v2, v1u3 2 v3u1, u1v2 2 u2v1d

2-12

MATHEMATICS

The cross product u  v is a three-dimensional vector that is perpendicular to both u and v. The cross product is not commutative. In fact, u 3 v 5 2v 3 u

¢

Cross product and inner product have two properties involving trigonometric functions. If u is the angle between vectors u and v, then uv 5 |u | |v| cos u

|u 3 v | 5 |u| |v | sin u

and

Matrices

A matrix is a rectangular array of numbers. A matrix A with m rows and n columns may be written a11 a12 a13 c a1n a21 a22 a23 c a2n A 5 • a31 a32 a33 c a3n µ c c c c c am1 am2 am3 c amn The numbers aij are called the entries of the matrix. The first subscript i identifies the row of the entry, and the second subscript j identifies the column. Matrices are denoted either by capital letters, A, B, etc., or by writing the general entry in parentheses, (aij ). The number of rows and the number of columns together define the dimensions of the matrix. The matrix A is an m  n matrix, read “m by n.” A row vector may be considered to be a 1  n matrix, and a column vector may be considered as a n  1 matrix. The rows of a matrix are sometimes considered as row vectors, and the columns may be considered as column vectors. If a matrix has the same number of rows as columns, the matrix is called a square matrix. In a square matrix, the entries aii, where the row index is the same as the column index, are called the diagonal entries. If a matrix has all its entries equal to zero, it is called a zero matrix. If a square matrix has all its entries equal to zero except its diagonal entries, it is called a diagonal matrix. The diagonal matrix with all its diagonal entries equal to 1 is called the identity matrix, and is denoted I, or Inn if it is important to emphasize the dimensions of the matrix. The 2  2 and 3  3 identity matrices are: I232 5 ¢

1 0 ≤ 0 1

1 0 0 I333 5 £0 1 0≥ 0 0 1

The entries of a square matrix aij where i  j are said to be below the diagonal. Similarly, those where i j are said to be above the diagonal. A square matrix with all entries below (resp. above) the diagonal equal to zero is called upper-triangular (resp. lower-triangular). Matrix Addition Matrices A and B may be added only if they have the same dimensions. Then the sum C  A  B is defined by cij 5 aij 1 bij That is, corresponding entries of the matrices are added together, just as with vectors. Similarly, matrices may be multiplied by scalars. Matrix Multiplication Matrices A and B may be multiplied only if the number of columns in A equals the number of rows of B. If A is an m  n matrix and B is an n  p matrix, then the product C  AB is an m  p matrix, defined as follows: c 5a b 1a b 1# # #1a b ij

i1 1j

i2 2j

EXAMPLE.

¢

1 2 3 4 ≤ ¢ ≤ 5 5 6 7 8

1331237 1341238 17 20 ≤ 5 ¢ ≤ 5331637 5341638 57 68

Matrix multiplication is not commutative. Even if A and B are both square, it is hardly ever true that AB  BA. Matrix multiplication does have the following properties: 1. (AB)C  A(BC) associative law 2. AsB 1 Cd 5 AB 1 AC r distributive laws 3. sB 1 CdA 5 BA 1 CA If A is square, then also 4. AI  IA  A multiplicative identity If A is square, then powers of A, AA, and AAA are denoted A2 and A3, respectively. The transpose of a matrix A, written AT, is obtained by writing the rows of A as columns. If A is m  n, then AT is n  m. EXAMPLE.

¢

T 1 4 1 2 3 ≤ 5 £2 5≥ 4 5 6 3 6

The transpose has the following properties: 1. (AT )T  A 2. (A  B)T  AT  BT 3. (AB)T  BTAT Note that in property 3, the order of multiplication is reversed. If AT  A, then A is called symmetric. Linear Equations

A linear equation in two variables is of the form a1x1 1 a2x2 5 b

or

a1x 1 a2 y 5 b

depending on whether the variables are named x1 and x2 or x and y. In n variables, such an equation has the form a1x1 1 a2x2 1 # # # anxn 5 b Such equations describe lines and planes. Often it is necessary to solve several such equations simultaneously. A set of m linear equations in n variables is called an m  n system of simultaneous linear equations. Systems with Two Variables 1  2 Systems An equation of the form

a1x 1 a2y 5 b has infinitely many solutions which form a straight line in the xy plane. That line has slope a1 /a2 and y intercept b/a2. 2  2 Systems A 2  2 system has the form a11x 1 a12y 5 b1

a21x 1 a22y 5 b2

Solutions to such systems do not always exist. CASE 1. The system has exactly one solution (Fig. 2.1.55a). The lines corresponding to the equations intersect at a single point. This occurs whenever the two lines have different slopes, so they are not

in nj

n

5 g aikbkj k51

The entry cij may also be defined as the dot product of row i of A with the transpose of column j of B.

Fig. 2.1.55 Lines corresponding to linear equations. (a) One solution; (b) no solutions; (c) infinitely many solutions.

LINEAR ALGEBRA

on the ij entry. Combining pivoting, the properties of the elementary row operations, and the fact:

parallel. In this case, a11 a12 a21 2 a22

2-13

a11a22 2 a21a12 2 0

so

CASE 2. The system has no solutions (Fig. 2.1.55b). This occurs whenever the two lines have the same slope and different y intercepts, so they are parallel. In this case,

|In3n| 5 1 provides a technique for finding the determinant of n  n matrices. EXAMPLE.

Find |A| where

a11 a12 a21 5 a22

1 2 24 A 5 £5 23 27≥ 3 22 3

CASE 3. The system has infinitely many solutions (Fig. 2.1.55c). This occurs whenever the two lines coincide. They have the same slope and y intercept. In this case,

First, pivot on the entry in row 1, column 1, in this case, the 1. Multiplying row 1 by 5, then adding row 1 to row 2, we first multiply the determinant by 5, then do not change it:

a11 a12 b1 a21 5 a22 5 b2 The value a11 a22  a21 a12 is called the determinant of the system. A larger n  n system also has a determinant (see below). A system has exactly one solution when its determinant is not zero. 3  2 Systems Any system with more equations than variables is called overdetermined. The only case in which a 3  2 system has exactly one solution is when one of the equations can be derived from the other two. One basic way to solve such a system is to treat any two equations as a 2  2 system and see if the solution to that subsystem of equations is also a solution to the third equation. Matrix Form for Systems of Equations The 2  2 system of linear equations a11x1 1 a12x2 5 b1

a21x1 1 a22x2 5 b2

25|A| 5 3

Next, multiply row 1 by 3⁄5 and add row 1 to row 3: 23|A| 5 3

or as

a11 a12 b x ≤ ¢ 1≤ 5 ¢ 1≤ a21 a22 x2 b2

det A 5 a11a22 2 a21a12 In general, any m  n system of simultaneous linear equations may be written as Ax 5 b where A is an m  n matrix, x is an n-dimensional column vector, and b is an m-dimensional column vector. An n  n (square) system of simultaneous linear equations has exactly one solution whenever its determinant is not zero. Then the system and the matrix A are called nonsingular. If the determinant is zero, the system is called singular. Elementary Row Operations on a Matrix There are three operations on a matrix which change the matrix: 1. Multiply each entry in row i by a scalar k (not zero). 2. Interchange row i with row j. 3. Add row i to row j. Similarly, there are three elementary column operations. The elementary row operations have the following effects on |A|: 1. Multiplying a row (or column) by k multiplies |A| by k. 2. Interchanging two rows (or columns) multiplies |A| by 1. 3. Adding one row (or column) to another does not change |A|. Pivoting, or Reducing, a Column The process of changing the ij entry of a matrix to 1 and changing the rest of column j to zero, by using elementary row operations, is known as reducing column j or as pivoting

26 12 23 26 12 213 13 3 5 3 0 213 13 3 22 3 0 28 15

1 |A| 5 3 0 0

2 24 213 13 3 2 8 15

Next, pivot on the entry in row 2, column 2. Multiplying row 2 by 8⁄13 and then adding row 2 to row 3, we get: 2

1 2 24 1 2 24 8 |A| 5 3 0 8 28 3 5 3 0 8 28 3 13 0 28 15 0 0 7

Next, divide row 2 by 8⁄13.

Ax  b

where A is the 2  2 matrix and x and b are two-dimensional column vectors. Then, the determinant of A, written det A or |A|, is the same as the determinant of the 2  2 system:

23 0 3

Next, divide row 1 by 3:

may be written as a matrix equation as follows: ¢

25 210 20 25 210 20 5 23 27 3 5 3 0 213 13 3 3 22 3 3 22 3

1 2 24 |A| 5 3 0 213 13 3 0 0 7 The determinant of a triangular matrix is the product of its diagonal elements, in this case 91. Inverses Whenever |A| is not zero, that is, whenever A is nonsingular, then there is another n  n matrix, denoted A1, read “A inverse” with the property

AA21 5 A21A 5 In3n Then the n  n system of equations Ax 5 b can be solved by multiplying both sides by A1, so x 5 In 3 nx 5 A21Ax 5 A21b x 5 A21b

so

The matrix A1 may be found as follows: 1. Make a n  2n matrix, with the first n columns the matrix A and the last n columns the identity matrix Inn. 2. Pivot on each of the diagonal entries of this matrix, one after another, using the elementary row operations. 3. After pivoting n times, the matrix will have in the first n columns the identity matrix, and the last n columns will be the matrix A1. EXAMPLE.

Solve the system x1 1 2x2 2 4x3 5 24 5x1 2 3x2 2 7x3 5 6 3x1 2 2x2 1 3x3 5 11

2-14

MATHEMATICS

A nonzero vector v satisfying

We must invert the matrix 1 2 24 A 5 £5 23 27≥ 3 22 3

sA 2 xiI dv 5 0

This is the same matrix used in the determinant example above. Adjoin the identity matrix to make a 3  6 matrix 1 2 24 1 £5 23 27 0 3 22 3 0 Perform the elementary row operations in determinant example. STEP 1.

Pivot on row 1, column 1. 1 £0 0

STEP 2.

24 13 15

2 213 28

1 25 23

0 1 0

0 0≥ 1

Pivot on row 2, column 2. 1 0 22 £0 1 21 0 0 7

STEP 3.

0 0 1 0≥ 0 1 exactly the same order as in the

3⁄13

2⁄13 0 21⁄13 0≥ 28⁄13 1

5⁄13 1⁄13

Pivot on row 3, column 3. 1 £0 0

0 1 0

23

⁄91 36⁄91

0 0 1

1⁄91

22⁄91 215⁄91 28⁄91

26⁄91 13⁄91≥ 13⁄91

Now, the inverse matrix appears on the right. To solve the equation,

is called an eigenvector of A associated with the eigenvalue xi. Eigenvectors have the special property Av 5 xiv Any multiple of an eigenvector is also an eigenvector. A matrix is nonsingular when none of its eigenvalues are zero. Rank and Nullity It is possible that the product of a nonzero matrix A and a nonzero vector v is zero. This cannot happen if A is nonsingular. The set of all vectors which become zero when multiplied by A is called the kernel of A. The nullity of A is the dimension of the kernel. It is a measure of how singular a matrix is. If A is an m  n matrix, then the rank of A is defined as n  nullity. Rank is at most m. The technique of pivoting is useful in finding the rank of a matrix. The procedure is as follows: 1. Pivot on each diagonal entry in the matrix, starting with a11. 2. If a row becomes all zero, exchange it with other rows to move it to the bottom of the matrix. 3. If a diagonal entry is zero but the row is not all zero, exchange the column containing the entry with a column to the right not containing a zero in that row. When the procedure can be carried no further, the nullity is the number of rows of zeros in the matrix. EXAMPLE.

Find the rank and nullity of the 3  2 matrix: 1 £2 4

x 5 A21b 23⁄91

so,

x 5 £36⁄91 1⁄91

22⁄91 215⁄91 28⁄91

24 13⁄91≥ £ 6≥ 13⁄91 11 26⁄91

Pivoting on row 1, column 1, yields

s24 3 23 1 6 3 22 1 11 3 26d/91 2 5 £ s24 3 36 1 6 3 215 1 11 3 13d/91≥ 5 £21≥ s24 3 1 1 6 3 28 1 11 3 13d/91 1 The solution to the system is then x1 5 2

1 £0 0

x3 5 1

possible to take the complex conjugate aij* of each entry, aij. This is called the conjugate of A and is denoted A*. 1. If aij  aji, then A is symmetric. 2. If aij  aji, then A is skew or antisymmetric. 3. If AT  A1, then A is orthogonal. 4. If A  A1, then A is involutory. 5. If A  A*, then A is hermitian. 6. If A  A*, then A is skew hermitian. 7. If A1  A*, then A is unitary. Eigenvalues and Eigenvectors If A is a square matrix and x is a variable, then the matrix B  A  xI is the characteristic matrix, or eigenmatrix, of A. The determinant |A  xI | is a polynomial of degree n, called the characteristic polynomial of A. The roots of this polynomial, x1, x2, . . . , xn, are the eigenvalues of A. Note that some sources define the characteristic matrix as xI  A. If n is odd, then this multiplies the characteristic equation by 1, but the eigenvalues are not changed. A5 2

22 5 2 2 1

B5 2

22 2 x 5 2 2 12x

Then the characteristic polynomial is

1 £0 0

0 1≥ 0

Nullity is therefore 1. Rank is 3  1  2.

If the rank of a matrix is n, so that Rank  nullity  m the matrix is said to be full rank. TRIGONOMETRY Formal Trigonometry Angles or Rotations An angle is generated by the rotation of a ray, as Ox, about a fixed point O in the plane. Every angle has an initial line (OA) from which the rotation started (Fig. 2.1.56), and a terminal line (OB) where it stopped; and the counterclockwise direction of rotation is taken as positive. Since the rotating ray may revolve as often as desired, angles of any magnitude, positive or negative, may be obtained. Two angles are congruent if they may be superimposed so that their initial lines coincide and their terminal lines coincide; i.e., two congruent angles are either equal or differ by some multiple of 360. Two angles are complementary if their sum is 90; supplementary if their sum is 180.

|B| 5 s22 2 xds1 2 xd 2 s2ds5d 5 x 2 1 x 2 2 2 10 5 x 2 1 x 2 12 5 sx 1 4dsx 2 3d The eigenvalues are 4 and 3.

0 21≥ 23

Pivoting on row 2, column 2, yields x2 5 21

Special Matrices If A is a matrix of complex numbers, then it is

EXAMPLE.

1 1≥ 1

Fig. 2.1.56 Angle.

TRIGONOMETRY

2-15

(The acute angles of a right-angled triangle are complementary.) If the initial line is placed so that it runs horizontally to the right, as in Fig. 2.1.57, then the angle is said to be an angle in the 1st, 2nd, 3rd, or 4th quadrant according as the terminal line lies across the region marked I, II, III, or IV.

perpendicular from P on OA or OA produced. In the right triangle OMP, the three sides are MP  “side opposite” O (positive if running upward); OM  “side adjacent” to O (positive if running to the right); OP  “hypotenuse” or “radius” (may always be taken as positive); and the six ratios between these sides are the principal trigonometric

Fig. 2.1.57 Circle showing quadrants.

Fig. 2.1.58 Unit circle showing elements used in trigonometric functions.

Units of Angular Measurement

1. Sexagesimal measure. (360 degrees  1 revolution.) Denoted on many calculators by DEG. 1 degree  1  1⁄90 of a right angle. The degree is usually divided into 60 equal parts called minutes (), and each minute into 60 equal parts called seconds (); while the second is subdivided decimally. But for many purposes it is more convenient to divide the degree itself into decimal parts, thus avoiding the use of minutes and seconds. 2. Centesimal measure. Used chiefly in France. Denoted on calculators by GRAD. (400 grades  1 revolution.) 1 grade  1⁄100 of a right angle. The grade is always divided decimally, the following terms being sometimes used: 1 “centesimal minute”  1⁄100 of a grade; 1 “centesimal second”  1⁄100 of a centesimal minute. In reading Continental books it is important to notice carefully which system is employed. 3. Radian, or circular, measure. (p radians  180 degrees.) Denoted by RAD. 1 radian  the angle subtended by an arc whose length is equal to the length of the radius. The radian is constantly used in higher mathematics and in mechanics, and is always divided decimally. Many theorems in calculus assume that angles are being measured in radians, not degrees, and are not true without that assumption. 1 radian  578.30  578.2957795131  5781744 s .806247  1808>p 18  0.01745 . . . radian  0.01745 32925 radian 1  0.00029 08882 radian 1 s  0.00000 48481 radian Table 2.1.2

sine of x 5 sin x 5 opp/hyp 5 MP/OP cosine of x 5 cos x 5 adj/hyp 5 OM/OP tangent of x 5 tan x 5 opp/adj 5 MP/OM cotangent of x 5 cot x 5 adj/opp 5 OM/MP secant of x 5 sec x 5 hyp/adj 5 OP/OM cosecant of x 5 csc x 5 hyp/opp 5 OP/MP The last three are best remembered as the reciprocals of the first three: cot x 5 1/ tan x

sec x 5 1/ cos x

csc x 5 1/ sin x

Trigonometric functions, the exponential functions, and complex numbers are all related by the Euler formula: eix  cos x  i sin x, where i 5 221. A special case of this ei p  1. Note that here x must be measured in radians. Variations in the functions as x varies from 0 to 360 are shown in Table 2.1.3. The variations in the sine and cosine are best remembered by noting the changes in the lines MP and OM (Fig. 2.1.59) in the “unit circle” (i.e., a circle with radius  OP  1), as P moves around the circumference.

Signs of the Trigonometric Functions

If x is in quadrant sin x and csc x are cos x and sec x are tan x and cot x are

I

II

III

IV

  

  

  

  

Definitions of the Trigonometric Functions Let x be any angle whose initial line is OA and terminal line OP (see Fig. 2.1.58). Drop a

Table 2.1.3

functions of the angle x; thus:

Fig. 2.1.59 Unit circle showing angles in the various quadrants.

Ranges of the Trigonometric Functions Values at

x in DEG x in RAD

08 to 908 (0 to p/2)

908 to 1808 (p/2 to p)

1808 to 2708 (p to 3p/2)

2708 to 3608 (3p/2 to 2p)

308 (p/6)

458 (p/4)

608 (p/3)

sin x csc x

0 to 1  ` to 1

1 to 0 1 to  `

0 to 1  ` to 1

1 to 0 1 to  `

1⁄2

1⁄2 !2

1⁄2 !3

cos x sec x

1 to 0 1 to  `

0 to 1  ` to 1

1 to 0 1 to  `

0 to 1  ` to 1

tan x cot x

0 to  `  ` to 0

 ` to 0 0 to  `

0 to  `  ` to 0

 ` to 0 0 to  `

2

1⁄2 !3 2

⁄3 !3

1⁄2 !3

!3

!2

1⁄2 !2

!2 1 1

⁄3 !3

2

1⁄2

2

!3 1⁄3 !3

2-16

MATHEMATICS

To Find Any Function of a Given Angle (Reduction to the first quadrant.) It is often required to find the functions of any angle x from a table that includes only angles between 0 and 90. If x is not already between 0 and 360, first “reduce to the first revolution” by simply adding or subtracting the proper multiple of 360 [for any function of (x)  the same function of (x n  360)]. Next reduce to first quadrant per table below.

90 and 180 (p/2 and p)

If x is between

180 from x (p)

270 from x (3p/2)

 sin (x  180)  csc (x  180)  cos (x  180)  sec (x  180)  tan (x  180)  cot (x  180)

 cos (x  270)  sec (x  270)  sin (x  270)  csc (x  270)  cot (x  270)  tan (x  270)

NOTE. The formulas for sine and cosine are best remembered by aid of the unit circle. To Find the Angle When One of Its Functions Is Given In general, there will be two angles between 0 and 360 corresponding to any given function. The rules showing how to find these angles are tabulated below. First find an acute angle x0 such that

sin x   a cos x   a tan x   a cot x   a

sin x0  a cos x0  a tan x0  a cot x0  a

sin x  a cos x  a tan x  a cot x  a

sin x0  a cos x0  a tan x0  a cot x0  a

Then the required angles x1 and x2 will be* x0 x0 x0 x0

and 180  x0 and [360  x0] and [180  x0] and [180  x0]

[180  x0] and [360  x0] 180  x0 and [180  x0] 180  x0 and [360  x0] 180  x0 and [360  x0]

* The angles enclosed in brackets lie outside the range 0 to 180 deg and hence cannot occur as angles in a triangle.

Relations Among the Functions of a Single Angle

sin2 x 1 cos2 x 5 1 sin x tan x 5 cos x cos x 1 5 tan x sin x 1 2 2 1 1 tan x 5 sec x 5 cos2 x 1 1 1 cot 2 x 5 csc 2 x 5 sin2 x tan x 1 sin x 5 21 2 cos2 x 5 5 21 1 tan2 x 21 1 cot 2 x cot x 1 cos x 5 21 2 sin2 x 5 5 21 1 tan2 x 21 1 cot 2 x cot x 5

sin x  sin y  2 cos 1⁄2(x  y) sin 1⁄2(x  y) cos x  cos y  2 cos 1⁄2(x  y) cos 1⁄2(x  y) cos x  cos y  2 sin 1⁄2(x  y) sin 1⁄2(x  y) sin sx 1 yd sin sx 1 yd tan x 1 tan y 5 cos x cos y ; cot x 1 cot y 5 sin x sin y sin sx 2 yd sin sx 2 yd tan x 2 tan y 5 cos x cos y ; cot x 2 cot y 5 sin x sin y sin2 x  sin2 y  cos2 y  cos2 x  sin (x  y) sin (x  y) cos2 x  sin2 y  cos2 y  sin2 x  cos (x  y) cos (x  y) sin (45  x)  cos (45  x) tan (45  x)  cot (45  x) sin (45  x)  cos (45  x) tan (45  x)  cot (45  x) In the following transformations, a and b are supposed to be positive, c 5 2a 2 1 b 2, A  the positive acute angle for which tan A  a/b, and B  the positive acute angle for which tan B  b/a: a cos x  b sin x  c sin (A  x)  c cos (B  x) a cos x  b sin x  c sin (A  x)  c cos (B  x) Functions of Multiple Angles and Half Angles

sin 2x  2 sin x cos x; sin x  2 sin 1⁄2x cos 1⁄2x cos 2x  cos2 x  sin2 x  1  2 sin2 x  2 cos2 x  1 cot 2 x 2 1 2 cot x 3 tan x 2 tan3 x sin 3x 5 3 sin x 2 4 sin3 x; tan 3x 5 1 2 3 tan2 x cos 3x  4 cos3 x  3 cos x sin snxd 5 n sin x cos n21 x 2 snd3 sin3 x cos n23 x 1snd5 sin5 x cosn25x 2 # # # n 2 n22 cos snxd 5 cos x 2 snd2 sin x cos x 1 snd4 sin4 x cosn24 x 2 # # # tan 2x 5

2 tan x 1 2 tan2 x

Functions of the Sum and Difference of Two Angles

cot 2x 5

where (n)2, (n)3, . . . , are the binomial coefficients. sin 1⁄2x 5 6 21⁄2 s1 2 cos xd. 1 2 cos x 5 2 sin2 1⁄2 x cos 1⁄2x 5 6 21⁄2 s1 1 cos xd. 1 1 cos x 5 2 cos2 1⁄2 x tan 1⁄2x 5 6

Functions of Negative Angles sin (x)  sin x; cos (x)  cos x;

tan (x)  tan x.

sin (x  y)  sin x cos y  cos x sin y cos (x  y)  cos x cos y  sin x sin y

270 and 360 (3p/2 and 2p)

90 from x (p/2)

The “reduced angle” (x  90, or x  180, or x  270) will in each case be an angle between 0 and 90, whose functions can then be found in the table.

Given

180 and 270 (p and 3p/2)

 cos (x  90)  sec (x  90)  sin (x  90)  csc (x  90)  cot (x  90)  tan (x  90)

Subtract Then sin x csc x cos x sec x tan x cot x

tan (x  y)  (tan x  tan y)/(1  tan x tan y) cot (x  y)  (cot x cot y  1)/(cot x  cot y) sin (x  y)  sin x cos y  cos x sin y cos (x  y)  cos x cos y  sin x sin y tan (x  y)  (tan x  tan y)/(1  tan x tan y) cot (x  y)  (cot x cot y  1)/(cot y  cot x) sin x  sin y  2 sin 1⁄2(x  y) cos 1⁄2(x  y)

tan ¢

1 2 cos x sin x 1 2 cos x 5 5 Å 1 1 cos x 1 1 cos x sin x

x 1 1 sin x 1 458≤ 5 6 Å 1 2 sin x 2

Here the  or  sign is to be used according to the sign of the lefthand side of the equation.

TRIGONOMETRY Approximations for sin x, cos x, and tan x For small values of x, x measured in radians, the following approximations hold:

sin x < x

tan x < x

cos x < 1 2

sin x , x , tan x

cos x ,

To find the remaining sides, use b5

x2 2

The following actually hold: sin x x ,1

As x approaches 0, lim [(sin x)/x]  1. Inverse Trigonometric Functions The notation sin1 x (read: arcsine of x, or inverse sine of x) means the principal angle whose sine is x. Similarly for cos1 x, tan1 x, etc. (The principal angle means an angle between 90 and 90 in case of sin1 and tan1, and between 0 and 180 in the case of cos1.)

a sin B sin A

c5

a sin C sin A

Or, drop a perpendicular from either B or C on the opposite side, and solve by right triangles. Check: c cos B  b cos C  a. CASE 2. GIVEN TWO SIDES (say a and b) AND THE INCLUDED ANGLE (C); AND SUPPOSE a  b (Fig. 2.1.63). Method 1: Find c from c2  a2  b2  2ab cos C; then find the smaller angle, B, from sin B  (b/c) sin C; and finally, find A from A  180  (B  C). Check: a cos B  b cos A  c. Method 2: Find 1⁄2(A  B) from the law of tangents: tan 1⁄2 sA 2 Bd 5 [sa 2 bd/sa 1 bd cot 1⁄2C

Solution of Plane Triangles

The “parts” of a plane triangle are its three sides a, b, c, and its three angles A, B, C (A being opposite a). Two triangles are congruent if all their corresponding parts are equal. Two triangles are similar if their corresponding angles are equal, that is, A1  A2, B1  B2, and C1  C2. Similar triangles may differ in scale, but they satisfy a1/a2  b1/b2  c1/c2. Two different triangles may have two corresponding sides and the angle opposite one of those sides equal (Fig. 2.1.60), and still not be congruent. This is the angle-side-side theorem. Otherwise, a triangle is uniquely determined by any three of its parts, as long as those parts are not all angles. To “solve” a triangle means to find the unknown parts from the known. The fundamental formulas are Law of sines:

2-17

and 1⁄2(A  B) from 1⁄2(A  B)  90  C/2; hence A  1⁄2(A  B)  1⁄2(A  B) and B  1⁄2(A  B)  1⁄2(A  B).. Then find c from c  a sin C/sin A or c  b sin C/sin B. Check: a cos B  b cos A  c. Method 3: Drop a perpendicular from A to the opposite side, and solve by right triangles. CASE 3. GIVEN THE THREE SIDES (provided the largest is less than the sum of the other two) (Fig. 2.1.64). Method 1: Find the largest angle A (which may be acute or obtuse) from cos A  (b2  c2  a2)/2bc and then find B and C (which will always be acute) from sin B  b sin A/a and sin C  c sin A/a. Check: A  B  C  180.

a sin A 5 b sin B

Law of cosines: c2 5 a 2 1 b 2 2 2ab cos C Fig. 2.1.63 Triangle with two sides and the included angle given.

Fig. 2.1.60 Triangles with an angle, an adjacent side, and an opposite side given. Right Triangles Use the definitions of the trigonometric functions, selecting for each unknown part a relation which connects that unknown with known quantities; then solve the resulting equations. Thus, in Fig. 2.1.61, if C  90, then A  B  90, c2  a2  b2,

sin A 5 a/c tan A 5 a/b

cos A 5 b/c cot A 5 b/a

If A is very small, use tan 1⁄2 A 5 2c 2 bd/sc 1 bd. Oblique Triangles There are four cases. It is highly desirable in all

these cases to draw a sketch of the triangle approximately to scale before commencing the computation, so that any large numerical error may be readily detected.

Fig. 2.1.61 Right triangle.

Fig. 2.1.62 Triangle with two angles and the included side given.

Fig. 2.1.64 Triangle with three sides given.

Method 2: Find A, B, and C from tan 1⁄2A  r/(s  a), tan 1⁄2B  r/(s  b), tan 1⁄2C  r/(s  c), where s  1⁄2(a  b  c), and r  2ss 2 adss 2 bdss 2 cd/s. Check: A  B  C  180. Method 3: If only one angle, say A, is required, use sin 1⁄2 A 5 2ss 2 bdss 2 cd/bc or

cos 1⁄2 A 5 2sss 2 ad/bc

according as 1⁄2 A is nearer 0 or nearer 90. CASE 4. GIVEN TWO SIDES (say b and c) AND THE ANGLE OPPOSITE ONE OF THEM (B). This is the “ambiguous case” in which there may be two solutions, or one, or none. First, try to find C  c sin B/b. If sin C  1, there is no solution. If sin C = 1, C  90 and the triangle is a right triangle. If sin C 1, this determines two angles C, namely, an acute angle C1, and an obtuse angle C2  180  C1. Then C1 will yield a solution when and only when C1  B 180 (see Case 1); and similarly C2 will yield a solution when and only when C2  B 180 (see Case 1). Other Properties of Triangles (See also Geometry, Areas, and Volumes.) Area  1⁄2ab sin C 5 2sss 2 adss 2 bdss 2 cd 5 rs where s  1⁄2(a  b  c), and r  radius of inscribed circle  2ss 2 adss 2 bdss 2 cd/s. Radius of circumscribed circle  R, where 2R 5 a/sin A 5 b/sin B 5 c/sin C

CASE 1. GIVEN TWO ANGLES (provided their sum is 180) AND ONE SIDE (say a, Fig. 2.1.62). The third angle is known since A  B  C  180.

C abc A B r 5 4R sin sin sin 5 2 2 2 4Rs

2-18

MATHEMATICS

The length of the bisector of the angle C is 2 2absss 2 cd 2ab[sa 1 bd 2 c ] 5 a1b a1b 2

z5

2

The median from C to the middle point of c is m  1 b 2d 2 c2.

closely related to the logarithmic function, and are especially valuable in the integral calculus. sinh21 sy/ad 5 ln sy 1 2y 2 1 a 2d 2 ln a cosh21 sy/ad 5 ln sy 1 2y 2 2 a 2d 2 ln a a1y y tanh21 a 5 1⁄2 ln a 2 y

1⁄2 22sa 2

Hyperbolic Functions

The hyperbolic sine, hyperbolic cosine, etc., of any number x, are functions of x which are closely related to the exponential ex, and which have formal properties very similar to those of the trigonometric functions, sine, cosine, etc. Their definitions and fundamental properties are as follows: sinh x 5 1⁄2 sex 2 e2xd cosh x 5 1⁄2 sex 1 e2xd tanh x 5 sinh x/cosh x cosh x 1 sinh x 5 ex cosh x 2 sinh x 5 e2x csch x 5 1/sinh x sech x 5 1/cosh x coth x 5 1/tanh x cosh2 x 2 sinh2 x 5 1 1 2 tanh2 x 5 sech2 x 1 2 coth2 x 5 2csch2 x

y1a y coth21 a 5 1⁄2 ln y 2 a ANALYTICAL GEOMETRY The Point and the Straight Line Rectangular Coordinates (Fig. 2.1.67) Let P1  (x1, y1), P2  (x2, y2).

Then, distance P1P2 5 2sx2 2 x1d2 1 sy2 2 y1d2 slope of P1 P2  m  tan u  (y2  y1)/(x2  x1); coordinates of midpoint are x  1⁄2(x1  x2), y  1⁄2(y1  y2); coordinates of point 1/nth of the way from P1 to P2 are x  x1  (1/n)(x2  x1), y  y1  (1/n)(y2  y1). Let m1, m2 be the slopes of two lines; then, if the lines are parallel, m1  m2; if the lines are perpendicular to each other, m1  1/m2.

sinh s2xd 5 2sinh x coshs2xd 5 cosh x tanh s2xd 5 2tanh x sinh sx 6 yd 5 sinh x cosh y 6 cosh x sinh y cosh sx 6 yd 5 cosh x cosh y 6 sinh x sinh y tanh sx 6 yd 5 stanh x 6 tanh yd/s1 6 tanh x tanh yd sinh 2x 5 2 sinh x cosh x cosh 2x 5 cosh2 x 1 sinh2 x tanh 2x 5 s2 tanh xd/s1 1 tanh2 xd sinh 1⁄2x 5 21⁄2 scosh x 2 1d cosh 1⁄2x 5 21⁄2 scosh x 1 1d tanh 1⁄2x 5 scosh x 2 1d/ssinh xd 5 ssinh xd/scosh x 1 1d

Fig. 2.1.68 Graph of straight line showing intercepts.

Fig. 2.1.67 Graph of straight line. Equations of a Straight Line

1. Intercept form (Fig. 2.1.68). x/a  y/b  1. (a, b  intercepts of the line on the axes.) 2. Slope form (Fig. 2.1.69). y  mx  b. (m  tan u  slope; b  intercept on the y axis.) 3. Normal form (Fig. 2.1.70). x cos v  y sin v  p. (p  perpendicular from origin to line; v  angle from the x axis to p.)

The hyperbolic functions are related to the rectangular hyperbola, x2  y2  a2 (Fig. 2.1.66), in much the same way that the trigonometric functions are related to the circle x2  y2  a2 (Fig. 2.1.65); the analogy, however, concerns not angles but areas. Thus, in either figure, let A Fig. 2.1.69 Graph of straight line showing slope and vertical intercept.

Fig. 2.1.70 Graph of straight line showing perpendicular line from origin.

4. Parallel-intercept form (Fig. 2.1.71). c  intercept on the parallel x  k).

Fig. 2.1.65 Circle.

y2b x 5 (b  y intercept, c2b k

Fig. 2.1.66 Hyperbola.

represent the shaded area, and let u  A/a2 (a pure number). Then for the coordinates of the point P we have, in Fig. 2.1.65, x  a cos u, y  a sin u; and in Fig. 2.1.66, x  a cosh u, y  a sinh u. The inverse hyperbolic sine of y, denoted by sinh1 y, is the number whose hyperbolic sine is y; that is, the notation x  sinh1 y means sinh x  y. Similarly for cosh1 y, tanh1 y, etc. These functions are

Fig. 2.1.71 Graph of straight line showing intercepts on parallel lines.

5. General form. Ax  By  C  0. [Here a  C/A, b  C/B, m  A/B, cos v  A/R, sin v  B/R, p  C/R, where R  6 2A2 1 B 2 (sign to be so chosen that p is positive).] 6. Line through (x1, y1) with slope m. y  y1  m(x  x1).

ANALYTICAL GEOMETRY

y 2 y1 7. Line through (x1, y1) and (x2, y2). y 2 y 5 2 1 x2 2 x1 sx 2 x1d. 8. Line parallel to x axis. y  a; to y axis: x  b.

Angles and Distances If u  angle from the line with slope m1 to

the line with slope m2, then tan u 5

sin u. For every value of the parameter u, there corresponds a point (x, y) on the circle. The ordinary equation x2  y2  a2 can be obtained from the parametric equations by eliminating u. The equation of a circle with radius a and center at (h, k) in parametric form will be x 5 h 1 a cos u; y 5 k 1 a sin u.

m2 2 m1 1 1 m2m1

If parallel, m1  m2. If perpendicular, m1m2  1. If u  angle between the lines Ax  By  C  0 and Ax  By  C  0, then AAr 1 BBr cos u 5 6 2sA2 1 B 2dsAr2 1 Br2d If parallel, A/A  B/B. If perpendicular, AA  BB  0. The equation of a line through (x1, y1) and meeting a given line y  mx  b at an angle u, is y 2 y1 5

2-19

m 1 tan u sx 2 x1d 1 2 m tan u

Fig. 2.1.73 Parameters of a circle. The Parabola

The parabola is the locus of a point which moves so that its distance from a fixed line (called the directrix) is always equal to its distance from a fixed point F (called the focus). See Fig. 2.1.74. The point halfway from focus to directrix is the vertex, O. The line through the focus, perpendicular to the directrix, is the principal axis. The breadth of the curve at the focus is called the latus rectum, or parameter,  2p, where p is the distance from focus to directrix.

The distance from (x0, y0) to the line Ax  By  C  0 is Ax0 1 By0 1 C

2 2A2 1 B 2 where the vertical bars mean “the absolute value of.” The distance from (x0, y0) to a line which passes through (x1, y1) and makes an angle u with the x axis is D5 2

D 5 sx0 2 x1d sin u 2 sy0 2 y1d cos u Polar Coordinates (Fig. 2.1.72) Let (x, y) be the rectangular and

(r, u) the polar coordinates of a given point P. Then x  r cos u; y  r sin u; x2  y2  r2.

Fig. 2.1.74 Graph of parabola. NOTE. Any section of a right circular cone made by a plane parallel to a tangent plane of the cone will be a parabola. Equation of parabola, principal axis along the x axis, origin at vertex

(Fig. 2.1.74): y2  2px.

Polar equation of parabola, referred to F as origin and Fx as axis (Fig. 2.1.75): r  p/(1  cos u). Equation of parabola with principal axis parallel to y axis: y  ax2  bx  c. This may be rewritten, using a technique called completing the

Fig. 2.1.72 Polar coordinates. Transformation of Coordinates If origin is moved to point (x0, y0), the new axes being parallel to the old, x  x0  x, y  y0  y. If axes are turned through the angle u, without change of origin,

x 5 xr cos u 2 yr sin u

y 5 xr sin u 1 yr cos u

square:

b2 b b2 y 5 a Bx 2 1 a x 1 2 R 1 c 2 4a 4a 2

5 a Bx 1

The Circle

b b2 R 1c2 2a 4a

The equation of a circle with center (a, b) and radius r is sx 2 ad2 1 sy 2 bd2 5 r 2 If center is at the origin, the equation becomes x2  y2  r 2. If circle goes through the origin and center is on the x axis at point (r, 0), equation becomes x2  y2  2rx. The general equation of a circle is x 2 1 y 2 1 Dx 1 Ey 1 F 5 0 It has center at (D/2, E/2), and radius 5 2sD/2d2 1 sE/2d2 2 F (which may be real, null, or imaginary). Equations of Circle in Parametric Form It is sometimes convenient to express the coordinates x and y of the moving point P (Fig. 2.1.73) in terms of an auxiliary variable, called a parameter. Thus, if the parameter be taken as the angle u from the x axis to the radius vector OP, then the equations of the circle in parametric form will be x  a cos u; y  a

Fig. 2.1.75 Polar plot of parabola.

Fig. 2.1.76 Vertical parabola showing rays passing through the focus.

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MATHEMATICS

Then: vertex is the point [b/2a, c  b2/4a]; latus rectum is p  1/2a; and focus is the point [b/2a, c  b2/4a  1/4a]. A parabola has the special property that lines parallel to its principal axis, when reflected off the inside “surface” of the parabola, will all pass through the focus (Fig. 2.1.76). This property makes parabolas useful in designing mirrors and antennas.

where v is the angle which the tangent at P makes with PF or PF. At end of major axis, R  b2/a  MA; at end of minor axis, R  a2/b  NB (see Fig. 2.1.81).

The Ellipse

The ellipse (as shown in Fig. 2.1.77), has two foci, F and F, and two directrices, DH and DH. If P is any point on the curve, PF  PF is constant,  2a; and PF/PH (or PF/PH) is also constant,  e, where e is the eccentricity (e 1). Either of these properties may be taken as the definition of the curve. The relations between e and the semiaxes a and b are as shown in Fig. 2.1.78. Thus, b 2 5 a 2 s1 2 e2d, ae 5 2a 2 2 b 2, e2  1  (b/a)2. The semilatus rectum  p  a(1  e2)  b2/a. Note that b is always less than a, except in the special case of the circle, in which b  a and e  0. Fig. 2.1.81 Ellipse showing radius of curvature. The Hyperbola

The hyperbola has two foci, F and F, at distances ae from the center, and two directrices, DH and DH, at distances a /e from the center (Fig. 2.1.82). If P is any point of the curve, | PF  PF| is constant,  2a; and PF/PH (or PF/PH) is also constant,  e (called the eccentricity), where e  1. Either of these properties may be taken as the Fig. 2.1.78 Ellipse showing semiaxes.

Fig. 2.1.77 Ellipse.

Any section of a right circular cone made by a plane which cuts all the elements of one nappe of the cone will be an ellipse; if the plane is perpendicular to the axis of the cone, the ellipse becomes a circle. Equation of ellipse, center at origin: y2 x2 1 251 2 a b

b y 5 6 a 2a 2 2 x 2

or

If P  (x, y) is any point of the curve, PF  a  ex, PF  a  ex. Equations of the ellipse in parametric form: x  a cos u, y  b sin u, where u is the eccentric angle of the point P  (x, y). See Fig. 2.1.81. Polar equation, focus as origin, axes as in Fig. 2.1.79. r  p/(1  e cos u). Equation of the tangent at (x1, y1): b2x1x  a2y1y  a2b2. The line y  mx  k will be a tangent if k 5 6 2a 2m 2 1 b 2.

Ellipse as a Flattened Circle, Eccentric Angle If the ordinates in a circle are diminished in a constant ratio, the resulting points will lie on an ellipse (Fig. 2.1.80). If Q traces the circle with uniform velocity, the corresponding point P will trace the ellipse, with varying velocity. The angle u in the figure is called the eccentric angle of the point P. A consequence of this property is that if a circle is drawn with its horizontal scale different from its vertical scale, it will appear to be an ellipse. This phenomenon is common in computer graphics. The radius of curvature of an ellipse at any point P  (x, y) is

R 5 a b sx /a 1 y /b d 2

4

definition of the curve. The curve has two branches which approach more and more nearly two straight lines called the asymptotes. Each asymptote makes with the principal axis an angle whose tangent is b/a. The relations between e, a, and b are shown in Fig. 2.1.83: b2  a2(e2  1), ae 5 2a 2 1 b 2, e2  1  (b/a)2. The semilatus rectum, or ordinate at the focus, is p  a(e2  1)  b2/a.

Fig. 2.1.80 Ellipse as a flattened circle.

Fig. 2.1.79 Ellipse in polar form.

2 2

Fig. 2.1.82 Hyperbola.

2

4 3/2

5 p/ sin v 3

Fig. 2.1.83 Hyperbola showing the asymptotes.

Any section of a right circular cone made by a plane which cuts both nappes of the cone will be a hyperbola. Equation of the hyperbola, center as origin: y2 x2 2 251 2 a b

or

b y 5 6 a 2x 2 2 a 2

ANALYTICAL GEOMETRY

2-21

If P  (x, y) is on the right-hand branch, PF  ex  a, PF  ex  a. If P is on the left-hand branch, PF  ex  a, PF  ex  a. Equations of Hyperbola in Parametric Form (1) x  a cosh u, y  b sinh u. Here u may be interpreted as A/ab, where A is the area shaded in Fig. 2.1.84. (2) x  a sec v, y  b tan v, where v is an auxiliary angle of no special geometric interest.

Fig. 2.1.87 Equilateral hyperbola.

Fig. 2.1.84 Hyperbola showing parametric form. Polar equation, referred to focus as origin, axes as in Fig. 2.1.85:

The length a  Th /w is called the parameter of the catenary, or the distance from the lowest point O to the directrix DQ (Fig. 2.1.89). When a is very large, the curve is very flat. The rectangular equation, referred to the lowest point as origin, is y  a [cosh (x/a)  1]. In case of very flat arcs (a large), y  x2/2a    ; s  x  1⁄6x3/a2    , approx, so that in such a case the catenary closely resembles a parabola.

r 5 p/s1 2 e cos ud Equation of tangent at (x1, y1): b2x1x  a2y1y  a2b2. The line y 

mx  k will be a tangent if k 5 6 2a 2m 2 2 b 2.

Fig. 2.1.88 Hyperbola with asymptotes as axes.

Fig. 2.1.85 Hyperbola in polar form.

The triangle bounded by the asymptotes and a variable tangent is of constant area,  ab. Conjugate hyperbolas are two hyperbolas having the same asymptotes with semiaxes interchanged (Fig. 2.1.86). The equations of the hyperbola conjugate to x2/a2  y2/b2  1 is x2/a2  y2/b2  1.

Calculus properties of the catenary are often discussed in texts on the calculus of variations (Weinstock, “Calculus of Variations,” Dover; Ewing, “Calculus of Variations with Applications,” Dover). Problems on the Catenary (Fig. 2.1.89) When any two of the four quantities, x, y, s, T/w are known, the remaining two, and also the parameter a, can be found, using the following: a 5 x/z T 5 wa cosh z s/x 5 ssinh zd/z

s 5 a sinh z y/x 5 scosh z 2 1d/z wx/T 5 z cosh z

Fig. 2.1.86 Conjugate hyperbolas. Equilateral Hyperbola (a  b) Equation referred to principal axes (Fig. 2.1.87): x2  y2  a2. NOTE. p  a (Fig. 2.1.87). Equation referred to asymptotes as axes (Fig. 2.1.88): xy  a2/2.

Asymptotes are perpendicular. Eccentricity  22. Any diameter is equal in length to its conjugate diameter.

Fig. 2.1.89 Catenary. NOTE. If wx/T 0.6627, then there are two values of z, one less than 1.2, and one greater. If wx/T  0.6627, then the problem has no solution.

The Catenary

Given the Length 2L of a Chain Supported at Two Points A and B Not in the Same Level, to Find a (See Fig. 2.1.90; b and c are supposed

The catenary is the curve in which a flexible chain or cord of uniform density will hang when supported by the two ends. Let w  weight of the chain per unit length; T  the tension at any point P; and Th, Tv  the horizontal and vertical components of T. The horizontal component Th is the same at all points of the curve.

NOTE. The coordinates of the midpoint M of AB (see Fig. 2.1.90) are x0  a tanh1 (b/L), y0  (L/tanh z)  a, so that the position of the lowest point is determined.

known.) Let s 2L2 2 b 2d/c 5 s/x; use s/x  sinh z/z to find z. Then a  c/z.

2-22

MATHEMATICS

Fig. 2.1.94). For the equations, put b  a in the equations of the epior hypotrochoid, below. Radius of curvature at any point P is R5 At A, R  0; at D, R 5

4asc 6 ad 3 sin 1⁄2u c 6 2a

4asc 6 ad . c 6 2a

Fig. 2.1.90 Catenary with ends at unequal levels. Other Useful Curves

The cycloid is traced by a point on the circumference of a circle which rolls without slipping along a straight line. Equations of cycloid, in parametric form (axes as in Fig. 2.1.91): x  a(rad u  sin u), y  a(1  cos u), where a is the radius of the rolling circle, and rad u is the radian measure of the angle u through which it has rolled. The radius of curvature at any point P is PC 5 4a sin su/2d 5 2 22ay.

Fig. 2.1.94 Hypocycloid. Special Cases If a  1⁄2c, the hypocycloid becomes a straight line, diameter of the fixed circle (Fig. 2.1.95). In this case the hypotrochoid traced by any point rigidly connected with the rolling circle (not necessarily on the circumference) will be an ellipse. If a  1⁄4c, the curve Fig. 2.1.91 Cycloid.

The trochoid is a more general curve, traced by any point on a radius of the rolling circle, at distance b from the center (Fig. 2.1.92). It is a prolate trochoid if b a, and a curtate or looped trochoid if b  a. The equations in either case are x  a rad u  b sin u, y  a  b cos u.

Fig. 2.1.92 Trochoid.

Fig. 2.1.95 Hypocycloid is straight line when the radius of inside circle is half that of the outside circle.

The epicycloid (or hypocycloid) is a curve generated by a point on the circumference of a circle of radius a which rolls without slipping on the outside (or inside) of a fixed circle of radius c (Fig. 2.1.93 and

generated will be the four-cusped hypocycloid, or astroid (Fig. 2.1.96), whose equation is x2/3  y2/3  c2/3. If a  c, the epicycloid is the cardioid, whose equation in polar coordinates (axes as in Fig. 2.1.97) is r  2c(1  cos u). Length of cardioid  16c. The epitrochoid (or hypotrochoid) is a curve traced by any point rigidly attached to a circle of radius a, at distance b from the center, when this

Fig. 2.1.93 Epicycloid.

Fig. 2.1.96 Astroid.

ANALYTICAL GEOMETRY

circle rolls without slipping on the outside (or inside) of a fixed circle of radius c. The equations are a a x 5 sc 6 ad cos ¢ c u ≤ 6 b cos B ¢1 6 c ≤u R a a y 5 sc 6 ad sin ¢ c u ≤ 2 b sin B ¢1 6 c ≤u R

2-23

v( angle POQ), are r  c sec v, rad u  tan v  rad v. Here, r  OP, and rad u  radian measure of angle, AOP (Fig. 2.1.98). The spiral of Archimedes (Fig. 2.1.99) is traced by a point P which, starting from O, moves with uniform velocity along a ray OP, while the ray itself revolves with uniform angular velocity about O. Polar equation: r  k rad u, or r  a(u/360). Here a  2pk  the distance measured along a radius, from each coil to the next. The radius of curvature at P is R  (k2  r2)3/2/(2k2  r2). The logarithmic spiral (Fig. 2.1.100) is a curve which cuts the radii from O at a constant angle v, whose cotangent is m. Polar equation: r  aem rad u. Here a is the value of r when u  0. For large negative values of u, the curve winds around O as an asymptotic point. If PT and PN are the tangent and normal at P, the line TON being perpendicular to OP (not shown in figure), then ON  rm, and PN 5 r 21 1 m 2 5 r/ sin v. Radius of curvature at P is PN.

Fig. 2.1.97 Cardioid.

where u  the angle which the moving radius makes with the line of centers; take the upper sign for the epi- and the lower for the hypotrochoid. The curve is called prolate or curtate according as b a or b  a. When b  a, the special case of the epi- or hypocycloid arises.

Fig. 2.1.100 Logarithmic spiral.

The tractrix, or Schiele’s antifriction curve (Fig. 2.1.101), is a curve such that the portion PT of the tangent between the point of contact and the x axis is constant  a. Its equation is

Fig. 2.1.98 Involute of circle.

The involute of a circle is the curve traced by the end of a taut string which is unwound from the circumference of a fixed circle, of radius c. If QP is the free portion of the string at any instant (Fig. 2.1.98), QP will be tangent to the circle at Q, and the length of QP  length of arc QA; hence the construction of the curve. The equations of the curve in parametric form (axes as in figure) are x  c(cos u  rad u sin u), y  c(sin u  rad u cos u), where rad u is the radian measure of the angle u which OQ makes with the x axis. Length of arc AP  1⁄2c(rad u)2; radius of curvature at P is QP. Polar equations, in terms of parameter

a x 5 6a B cosh21 y 2

Å

y 2 1 2 ¢a≤ R

or, in parametric form, x  a(t  tanh t), y  a/cosh t. The x axis is an asymptote of the curve. Length of arc BP  a loge (a /y).

Fig. 2.1.101 Tractrix.

The tractrix describes the path taken by an object being pulled by a string moving along the x axis, where the initial position of the object is B and the opposite end of the string begins at O.

Fig. 2.1.99 Spiral of Archimedes.

Fig. 2.1.102 Lemniscate.

2-24

MATHEMATICS

The lemniscate (Fig. 2.1.102) is the locus of a point P the product of whose distances from two fixed points F, F is constant, equal to 1⁄2a2. The distance FFr 5 a 22, Polar equation is r 5 a 2cos 2u. Angle between OP and the normal at P is 2u. The two branches of the curve cross at right angles at O. Maximum y occurs when u  30 and r 5 a/ 22, and is equal to 1⁄4a 22. Area of one loop  a2/2. The helix (Fig. 2.1.103) is the curve of a screw thread on a cylinder of radius r. The curve crosses the elements of the cylinder at a constant angle, v. The pitch, h, is the distance between two coils of the helix, measured along an element of the cylinder; hence h  2pr tan v. Length Fig. 2.1.103 Helix. of one coil  2s2prd2 1 h2 5 2pr> cos v. If the cylinder is rolled out on a plane, the development of the helix will be a straight line, with slope equal to tan v. DIFFERENTIAL AND INTEGRAL CALCULUS Derivatives and Differentials Derivatives and Differentials A function of a single variable x may be denoted by f(x), F(x), etc. The value of the function when x has the value x0 is then denoted by f(x0), F(x0), etc. The derivative of a function y  f(x) may be denoted by f (x), or by dy/dx. The value of the derivative at a given point x  x0 is the rate of change of the function at that point; or, if the function is represented by a curve in the usual way (Fig. 2.1.104), the value of the derivative at any point shows the slope of the curve (i.e., the slope of the tangent to the curve) at that point (positive if the tangent points upward, and negative if it points downward, moving to the right).

Fig. 2.1.104 Curve showing tangent and derivatives.

The increment y (read: “delta y”) in y is the change produced in y by increasing x from x0 to x0  x; i.e., y  f(x0  x)  f(x0). The differential, dy, of y is the value which y would have if the curve coincided with its tangent. (The differential, dx, of x is the same as x when x is the independent variable.) Note that the derivative depends only on the value of x0, while y and dy depend not only on x0 but on the value of x as well. The ratio y/x represents the secant slope, and dy/dx the slope of tangent (see Fig. 2.1.104). If x is made to approach zero, the secant approaches the tangent as a limiting position, so that the derivative is f rsxd 5

f sx0 1 xd 2 f sx0d dy y 5 lim B R 5 lim B R xS0 dx xS0 x x

Also, dy  f(x) dx. The symbol “lim” in connection with x S 0 means “the limit, as x approaches 0, of . . . .” (A constant c is said to be the limit of a variable

u if, whenever any quantity m has been assigned, there is a stage in the variation process beyond which |c  u| is always less than m; or, briefly, c is the limit of u if the difference between c and u can be made to become and remain as small as we please.) To find the derivative of a given function at a given point: (1) If the function is given only by a curve, measure graphically the slope of the tangent at the point in question; (2) if the function is given by a mathematical expression, use the following rules for differentiation. These rules give, directly, the differential, dy, in terms of dx; to find the derivative, dy/dx, divide through by dx. Rules for Differentiation (Here u, v, w, . . . represent any functions of a variable x, or may themselves be independent variables. a is a constant which does not change in value in the same discussion; e  2.71828.) 1. d(a  u)  du 2. d(au)  a du 3. d(u  v  w    )  du  dv  dw     4. d(uv)  u dv  v du dw dv du 5. dsuvwcd 5 suvwcd¢ u 1 v 1 w 1 c≤ v du 2 u dv u 6. d v 5 v2 7. d(um)  mum1 du. Thus, d(u2)  2u du; d(u3)  3u2 du; etc. du 8. d 2u 5 2 2u du 1 9. d ¢ u ≤ 5 2 2 u 10. d(eu)  eu du 11. d(au)  (ln a)au du 12. d ln u 5 du u 13. d log u 5 log e du 5 s0.4343 cd du 10 10 u u 14. d sin u  cos u du 15. d csc u  cot u csc u du 16. d cos u  sin u du 17. d sec u  tan u sec u du 18. d tan u  sec2 u du 19. d cot u  csc2 u du du 21 20. d sin u 5 21 2 u 2 du 21 21. d csc u 5 2 u 2u 2 2 1 du 21 22. d cos u 5 21 2 u 2 du 21 23. d sec u 5 u 2u 2 2 1 du 24. d tan21 u 5 1 1 u2 du 25. d cot 21 u 5 2 1 1 u2 26. d ln sin u  cot u du 2 du 27. d ln tan u 5 sin 2u 28. d ln cos u  tan u du 2 du 29. d ln cot u 5 2 sin 2u 30. d sinh u  cosh u du 31. d csch u  csch u coth u du 32. d cosh u  sinh u du 33. d sech u  sech u tanh u du

DIFFERENTIAL AND INTEGRAL CALCULUS

34. d tanh u  sech2 u du 35. d coth u  csch2 u du du 21 36. d sinh u 5 2u 2 1 1 du 21 37. d csch u 5 2 u 2u 2 1 1 du 21 38. d cosh u 5 2u 2 2 1 du 21 39. d sech u 5 2 u 21 2 u 2 du 40. d tanh21 u 5 1 2 u2 du 41. d coth21 u 5 1 2 u2 42. dsu vd 5 su v21dsu ln u dv 1 v dud Derivatives of Higher Orders The derivative of the derivative is called the second derivative; the derivative of this, the third derivative; and so on. If y  f(x), f rsxd 5 Dxy 5 f ssxd 5 D 2x y 5

dy dx d 2y dx 2 d 3y

2-25

If increments x, y (or dx, dy) are assigned to the independent variables x, y, the increment, u, produced in u  f (x, y) is u 5 f sx 1 x, y 1 yd 2 fsx, yd while the differential, du, i.e., the value which u would have if the partial derivatives of u with respect to x and y were constant, is given by du 5 s fxd # dx 1 s fyd # dy Here the coefficients of dx and dy are the values of the partial derivatives of u at the point in question. If x and y are functions of a third variable t, then the equation dy du dx 5 s fxd 1 s fyd dt dt dt expresses the rate of change of u with respect to t, in terms of the separate rate of change of x and y with respect to t. Implicit Functions If f(x, y)  0, either of the variables x and y is said to be an implicit function of the other. To find dy/dx, either (1) solve for y in terms of x, and then find dy/dx directly; or (2) differentiate the equation through as it stands, remembering that both x and y are variables, and then divide by dx; or (3) use the formula dy/dx  ( fx /fy), where fx and fy are the partial derivatives of f(x, y) at the point in question. Maxima and Minima

flection.

A function of one variable, as y  f(x), is said to have a maximum at a point x  x0, if at that point the slope of the curve is zero and the concavity downward (see Fig. 2.1.106); a sufficient condition for a maximum is f(x0)  0 and f (x0) negative. Similarly, f(x) has a minimum if the slope is zero and the concavity upward; a sufficient condition for a minimum is f (x 0)  0 and f (x 0) positive. If f (x0)  0 and f(x0)  0, the point x0 will be a point of inflection. If f(x0)  0 and f (x0)  0 and f (x 0)  0, the point x0 will be a maximum if f (x0) 0, and a minimum if f (x 0)  0. It is usually sufficient, however, in any practical case, to find the values of x which make f(x)  0, and then decide, from a general knowledge of the curve or the sign of f (x) to the right and left of x0, which of these values (if any) give maxima or minima, without investigating the higher derivatives.

Fig. 2.1.105 Curve showing concavity.

Fig. 2.1.106 Curve showing maxima and minima.

f - sxd 5 D 3x y 5

dx 3

etc.

NOTE. If the notation d 2y/dx2 is used, this must not be treated as a fraction, like dy/dx, but as an inseparable symbol, made up of a symbol of operation d 2/dx2, and an operand y.

The geometric meaning of the second derivative is this: if the original function y  f(x) is represented by a curve in the usual way, then at any point where f (x) is positive, the curve is concave upward, and at any point where f (x) is negative, the curve is concave downward (Fig. 2.1.105). When f (x)  0, the curve usually has a point of in-

Functions of two or more variables may be denoted by f (x, y, . . .), F(x, y, . . .), etc. The derivative of such a function u  f (x, y, . . .) formed on the assumption that x is the only variable (y, . . . being regarded for the moment as constants) is called the partial derivative of u with respect to x, and is denoted by fx (x, y) or Dxu, or dxu/dx, or 'u/'x. Similarly, the partial derivative of u with respect to y is fy(x, y) or Dyu, or dyu/dy, or 'u/'y. NOTE. In the third notation, dxu denotes the differential of u formed on the assumption that x is the only variable. If the fourth notation, 'u/'x, is used, this must not be treated as a fraction like du/dx; the '/'x is a symbol of operation, operating on u, and the “'x” must not be separated.

Partial derivatives of the second order are denoted by fxx, fxy, fyy, or by Du, Dx(Dyu), D 2y u, or by '2u/'x 2, '2u/'x 'y, '2u/'y 2, the last symbols being “inseparable.” Similarly for higher derivatives. Note that fxy  fyx.

A function of two variables, as u  f (x, y), will have a maximum at a point (x0, y0) if at that point fx  0, fy  0, and fxx 0, fyy 0; and a minimum if at that point fx  0, fy  0, and fxx  0, fyy  0; provided, in each case, ( fxx)( fyy)  ( fxy)2 is positive. If fx  0 and fy  0, and fxx and fyy have opposite signs, the point (x0, y0) will be a “saddle point” of the surface representing the function. Indeterminate Forms

In the following paragraphs, f(x), g(x) denote functions which approach 0; F(x), G(x) functions which increase indefinitely; and U(x) a function which approaches 1, when x approaches a definite quantity a. The problem in each case is to find the limit approached by certain combinations of these functions when x approaches a. The symbol S is to be read “approaches” or “tends to.” CASE 1. “0/0.” To find the limit of f(x)/g(x) when f(x) S 0 and g(x) S 0, use the theorem that lim [ f(x)/g(x)]  lim [ f (x)/g(x)],

2-26

MATHEMATICS

where f(x) and g(x) are the derivatives of f(x) and g(x). This second limit may be easier to find than the first. If f(x) S 0 and g(x) S 0, apply the same theorem a second time: lim [ f(x)/g(x)]  lim [ f (x)/ g(x)], and so on. CASE 2. “ ` / ` .” If F(x) S ` and G(x) S ` , then lim [F(x)/ G(x)]  lim [F(x)/G(x)], precisely as in Case 1. CASE 3. “0  ` .” To find the limit of f(x)  F(x) when f (x) S 0 and F(x) S ` , write lim [f(x)  F(x)]  lim{ f(x)/[1/F(x)]} or  lim {F(x)/ [1/f(x)]}, then proceed as in Case 1 or Case 2. CASE 4. The limit of combinations “00” or [f(x)]g(x); “1`” or [U(x)]F(x); “ ` 0” or [F(x)]b(x) may be found since their logarithms are limits of the type evaluated in Case 3. CASE 5. “ `  ` .” If F(x) S ` and G(x) S ` , write lim [Fsxd 2 Gsxd] 5 lim

1/Gsxd 2 1/Fsxd 1/[Fsxd # Gsxd]

then proceed as in Case 1. Sometimes it is shorter to expand the functions in series. It should be carefully noticed that expressions like 0/0, ` / ` , etc., do not represent mathematical quantities. Curvature

The radius of curvature R of a plane curve at any point P (Fig. 2.1.107) is the distance, measured along the normal, on the concave side of the curve, to the center of curvature, C, this point being the limiting position of the point of intersection of the normals at P and a neighboring point Q, as Q is made to approach P along the curve. If the equation of the curve is y  f(x), R5

[1 1 syrd2]3>2 ds 5 du ys

where ds  2dx 2 1 dy 2  the differential of arc, u  tan1 [ f(x)]  the angle which the tangent at P makes with the x axis, and y  f(x) and y  f (x) are the first and second derivatives of f (x) at the point P. Note that dx  ds cos u and dy  ds sin u. The curvature, K, at the point P, is K  1/R  du/ds; i.e., the curvature is the rate at which the angle u is changing with respect to the length of arc s. If the slope of the curve is small, K < f ssxd.

is easy. The most common integrable forms are collected in the following brief table; for a more extended list, see Peirce, “Table of Integrals,” Ginn, or Dwight, “Table of Integrals and other Mathematical Data,” Macmillan, or “CRC Mathematical Tables.” GENERAL FORMULAS 1. 3 a du 5 a 3 du 5 au 1 C 2. 3 su 1 vd dx 5 3 u dx 1 3v dx 3. 3 u dv 5 uv 2 3 v du

(integration by parts)

4. 3 f sxd dx 5 3 f [Fs yd]Fr syd dy, x 5 Fs yd (change of variables) 5. 3 dy 3 f sx, yd dx 5 3 dx 3 f sx, yd dy FUNDAMENTAL INTEGRALS x n11 6. 3 x n dx 5 1 C, when n 2 21 n11 dx 7. 3 x 5 ln x 1 C 5 ln cx 8. 3 ex dx 5 ex 1 C 9. 3 sin x dx 5 2cos x 1 C 10. 3 cos x dx 5 sin x 1 C dx 11. 3 5 3 csc 2 x dx 5 2 cot x 1 C sin 2 x dx 12. 3 5 3sec2 x dx 5 tan x 1 C cos 2 x dx 13. 3 5 sin 21 x 1 C 5 2 cos 21 x 1 C 21 2 x 2 dx 14. 3 5 tan 21 x 1 C 5 2 cot 21 x 1 C 1 1 x2 RATIONAL FUNCTIONS sa 1 bxdn11 15. 3 sa 1 bxdn dx 5 1C sn 1 1d b

Fig. 2.1.107 Curve showing radius of curvature.

If the equation of the curve in polar coordinates is r  f(u), where r  radius vector and u  polar angle, then R5

[r 2 1 srrd2]3>2 r 2 rrs 1 2srrd2 2

where r  f(u) and r  f (u). The evolute of a curve is the locus of its centers of curvature. If one curve is the evolute of another, the second is called the involute of the first. Indefinite Integrals

An integral of f(x) dx is any function whose differential is f(x) dx, and is denoted by f(x) dx. All the integrals of f(x) dx are included in the expression f(x) dx  C, where  f(x) dx is any particular integral, and C is an arbitrary constant. The process of finding (when possible) an integral of a given function consists in recognizing by inspection a function which, when differentiated, will produce the given function; or in transforming the given function into a form in which such recognition

dx 1 1 16. 3 5 ln sa 1 bxd 1 C 5 ln csa 1 bxd a 1 bx b b dx 1 1C 17. 3 n 5 2 except when n 5 1 x sn 2 1dx n21 dx 1 52 1C 18. 3 bsa 1 bxd sa 1 bxd2 dx 11x 19. 3 5 1⁄2 ln 1 C 5 tanh21 x 1 C, when x , 1 12x 1 2 x2 dx x21 20. 3 2 5 1⁄2 ln 1 C 5 2coth21 x 1 C, when x . 1 x11 x 21 dx b 1 5 tan21 a a x b 1 C 21. 3 Ä a 1 bx 2 2ab 2ab 1 bx dx 1 1C 5 ln 22. 3 a 2 bx 2 2 2ab 2ab 2 bx b 1 5 tanh21 a a x b 1 C Ä 2ab



[a . 0, b . 0]

DIFFERENTIAL AND INTEGRAL CALCULUS

dx 23. 3 5 a 1 2bx 1 cx 2 1

tan21

b 1 cx

1C

∂ [ac 2 b 2 . 0]

2ac 2 b 2 2ac 2 b 2 2 2b 2 ac 2 b 2 cx 1 5 ln 1C 2 2 2 2b 2 ac 2b 2 ac 1 b 1 cx ∂ [b 2 2 ac . 0] b 1 cx 1 tanh21 1C 52 2b 2 2 ac 2b 2 2 ac dx 1 24. 3 52 1 C, when b 2 5 ac b 1 cx a 1 2bx 1 cx 2 sm 1 nxd dx n 25. 3 5 ln sa 1 2bx 1 cx 2d 2c a 1 2bx 1 cx 2 dx mc 2 nb 1 3 a 1 2bx 1 cx 2 c fsxd dx 26. In 3 , if f(x) is a polynomial of higher than the first a 1 2bx 1 cx 2 degree, divide by the denominator before integrating dx 1 27. 3 5 sa 1 2bx 1 cx 2dp 2sac 2 b 2ds p 2 1d b 1 cx 3 sa 1 2bx 1 cx 2dp21 s2p 2 3dc dx 1 2sac 2 b 2ds p 2 1d 3 sa 1 2bx 1 cx 2dp21 sm 1 nxd dx n 28. 3 3 52 2cs p 2 1d sa 1 2bx 1 cx 2dp dx mc 2 nb 1 1 3 sa 1 2bx 1 cx 2dp c sa 1 2bx 1 cx 2d p21 x m21 sa 1 bxdn11 29. 3 x m21 sa 1 bxdn dx 5 sm 1 ndb sm 2 1da m22 2 x sa 1 bxdn dx sm 1 ndb 3 x m sa 1 bxdn na 5 1 x m21 sa 1 bxdn21 dx m1n m 1 n3

2 30. 3 2a 1 bx dx 5 s 2a 1 bxd3 1 C 3b dx 2 5 2a 1 bx 1 C 31. 3 b 2a 1 bx sm 1 nxd dx

33. 3

2a 1 bx dx

5

2 s3mb 2 2an 1 nbxd 2a 1 bx 1 C 3b 2

sm 1 nxd 2a 1 bx

; substitute y 5 2a 1 bx, and use 21 and 22

n

35. 36. 37. 38.

f sx, 2a 1 bxd

n

dx; substitute 2a 1 bx 5 y n Fsx, 2a 1 bxd dx x x 5 sin 21 a 1 C 5 2 cos 21 a 1 C 3 2a 2 2 x 2 dx x 5 ln sx 1 2a 2 1 x 2d 1 C 5 sinh21 a 1 C 3 2a 2 1 x 2 dx x 5 ln sx 1 2x 2 2 a 2d 1 C 5 cosh21 a 1 C 3 2 2 2x 2 a dx 3 2a 1 2bx 1 cx 2 1 5 ln sb 1 cx 1 2c 2a 1 2bx 1 cx 2d 1 C, where c . 0 2c

34. 3

5 5

1 2c 1 2c 21

sinh21 cosh

b 1 cx

2ac 2 b 2 21 b 1 cx 2b 2 2 ac b 1 cx 21

1 C, when ac 2 b 2 . 0 1 C, when b 2 2 ac . 0

sin 1 C, when c , 0 22c 2b 2 2 ac sm 1 nxd dx n 5 c 2a 1 2bx 1 cx 2 39. 3 2a 1 2bx 1 cx 2 mc 2 nb 1 3 c

dx

2a 1 2bx 1 cx 2 m m22 m21 sm 2 1da x x dx dx x X 5 mc 2 40. 3 3 X mc 2a 1 2bx 1 cx 2 s2m 2 1db x m21 2 2 3 X dx when X 5 2a 1 2bx 1 cx mc x a2 41. 3 2a 2 1 x 2 dx 5 2a 2 1 x 2 1 ln sx 1 2a 2 1 x 2d 1 C 2 2 x x a2 5 2a 2 1 x 2 1 sinh21 a 1 C 2 2 x x a2 42. 3 2a 2 2 x 2 dx 5 2a 2 2 x 2 1 sin 21 a 1 C 2 2 x a2 43. 3 2x 2 2 a 2 dx 5 2x 2 2 a 2 2 ln sx 1 2x 2 2 a 2d 1 C 2 2 x x a2 5 2x 2 2 a 2 2 cosh21 a 1 C 2 2 44. 3 2a 1 2bx 1 cx 2 dx 5

b 1 cx 2a 1 2bx 1 cx 2 2c dx ac 2 b 2 1C 1 2c 3 2a 1 2bx 1 cx 2

TRANSCENDENTAL FUNCTIONS ax 45. 3 a x dx 5 1C ln a 46. 3 x neax dx 5

IRRATIONAL FUNCTIONS

32. 3

5

2-27

nsn 2 1d x neax n # # # 6 n! d 1C a c1 2 ax 1 a 2x 2 2 a nx n

47. 3 ln x dx 5 x ln x 2 x 1 C ln x ln x 1 48. 3 2 dx 5 2 x 2 x 1 C x sln xdn 1 49. 3 x dx 5 sln xdn11 1 C n11 50. 3 sin 2 x dx 5 21⁄4 sin 2x 1 1⁄2 x 1 C 5 21⁄2 sin x cos x 1 1⁄2x 1 C 51. 3 cos 2 x dx 5 1⁄4 sin 2x 1 1⁄2x 1 C 5 1⁄2 sin x cos x 1 1⁄2x 1 C cos mx 52. 3 sin mx dx 5 2 m 1 C 53. 3 cos mx dx 5

sin mx m 1C

cos sm 1 ndx cos sm 2 ndx 54. 3 sin mx cos nx dx 5 2 2 1C 2sm 1 nd 2sm 2 nd sin sm 2 ndx sin sm 1 ndx 55. 3 sin mx sin nx dx 5 2 1C 2sm 2 nd 2sm 1 nd sin sm 2 ndx sin sm 1 ndx 56. 3 cos mx cos nx dx 5 1 1C 2sm 2 nd 2sm 1 nd

2-28

MATHEMATICS

dy A 1 B cos x 1 C sin x 77. 3 dx 5 A 3 a 1 p cos y a 1 b cos x 1 c sin x cos y dy 1 sB cos u 1 C sin ud 3 a 1 p cos y sin y dy 2 sB sin u 2 C cos ud 3 where b 5 p cos u, c 5 p a 1 p cos y’ sin u and x 2 u 5 y a sin bx 2 b cos bx ax 78. 3 eax sin bx dx 5 e 1C a2 1 b 2 a cos bx 1 b sin bx ax 79. 3 eax cos bx dx 5 e 1C a2 1 b 2

57. 3 tan x dx 5 2ln cos x 1 C 58. 3 cot x dx 5 ln sin x 1 C dx x 59. 3 5 ln tan 1 C 2 sin x dx x p 60. 3 cos x 5 ln tan a 1 b 1 C 4 2 dx x 61. 3 5 tan 1 C 1 1 cos x 2 dx x 62. 3 5 2 cot 1 C 1 2 cos x 2

80. 3 sin21 x dx 5 x sin21 x 1 21 2 x 2 1 C

63. 3 sin x cos x dx 5 1⁄2 sin x 1 C 2

dx 64. 3 5 ln tan x 1 C sin x cos x cos x sin n21 x n21 65.* 3 sin n x dx 5 2 1 n 3 sin n22 x dx n sin x cos n21 x n21 66.* 3 cos n x dx 5 1 n 3 cos n22 x dx n tan n21 x 67. 3 tan n x dx 5 2 3 tan n22 x dx n21 cot n21 x 68. 3 cot n x dx 5 2 2 3 cot n22 x dx n21 dx dx cos x n22 52 1 69. 3 n 2 1 3 sin n22 x sin n x sn 2 1d sin n21 x dx dx sin x n22 5 1 70. 3 cos n x n 2 1 3 cos n22 x sn 2 1d cos n21 x 71.† 3 sin p x cos q x dx 5

sin

p11

x cos p1q

q21

72.† 3 sin 2p x cos q x dx 5 2

73.† 3 sin p x cos 2q x dx 5

sin

sin

2q11

x cos x q21 q2p22 1 sin p x cos 2q12 x dx q21 3

dx a2b 2 74. 3 tan 21 a tan 1⁄2xb 1 C, 5 a 1 b cos x Äa 1 b 2a 2 2 b 2 when a 2 . b 2, b 1 a cos x 1 sin x 2b 2 2 a 2 1 5 ln 1 C, a 1 b cos x 2b 2 2 a 2 when a 2 , b 2,

b2a tanh21 a tan 1⁄2 xb 1 C, when a 2 , b 2 Äb 1 a 2b 2 a cos x dx dx x a 75. 3 5 2 3 1C a 1 b cos x b b a 1 b cos x sin x dx 1 76. 3 5 2 ln sa 1 b cos xd 1 C a 1 b cos x b 5

2

2

83. 3 cot 21 x dx 5 x cot 21 x 1 1⁄2 ln s1 1 x 2d 1 C 84. 3 sinh x dx 5 cosh x 1 C 85. 3 tanh x dx 5 ln cosh x 1 C 86. 3 cosh x dx 5 sinh x 1 C 87. 3 coth x dx 5 ln sinh x 1 C 88. 3 sech x dx 5 2 tan 21 sexd 1 C 89. 3 csch x dx 5 ln tanh sx/2d 1 C 90. 3 sinh2 x dx 5 1⁄2 sinh x cosh x 2 1⁄2 x 1 C 91. 3 cosh2 x dx 5 1⁄2 sinh x cosh x 1 1⁄2 x 1 C

q11

x cos x p21 p2q22 sin 2p12 x cos q x dx 1 p21 3

p11

82. 3 tan21 x dx 5 x tan21 x 2 1⁄2 ln s1 1 x 2d 1 C

x

q21 sin p21 x cos q11 x 1 sin p x cos q22 x dx 5 2 p 1 q3 p1q p21 sin p22 x cos q x dx 1 p 1 q3 2p11

81. 3 cos21 x dx 5 x cos21 x 2 21 2 x 2 1 C

2

* If n is an odd number, substitute cos x  z or sin x  z. † If p or q is an odd number, substitute cos x  z or sin x  z.

92. 3 sech2 x dx 5 tanh x 1 C 93. 3 csch2 x dx 5 2coth x 1 C Hints on Using Integral Tables It happens with frustrating frequency that no integral table lists the integral that needs to be evaluated. When this happens, one may (a) seek a more complete integral table, (b) appeal to mathematical software, such as Mathematica, Maple, MathCad or Derive, (c) use numerical or approximate methods, such as Simpson’s rule (see section “Numerical Methods”), or (d) attempt to transform the integral into one which may be evaluated. Some hints on such transformation follow. For a more complete list and more complete explanations, consult a calculus text, such as Thomas, “Calculus and Analytic Geometry,” Addison-Wesley, or Anton, “Calculus with Analytic Geometry,” Wiley. One or more of the following “tricks” may be successful.

TRIGONOMETRIC SUBSTITUTIONS 1. If an integrand contains 2sa 2 2 x 2d, substitute x  a sin u, and 2sa 2 2 x 2d  a cos u. 2. Substitute x  a tan u and 2sx 2 1 a 2d  a sec u. 3. Substitute x  a sec u and 2sx 2 2 a 2d  a tan u. COMPLETING THE SQUARE 4. Rewrite ax2  bx  c  a[x  b/(2a)]2  (4ac  b2)/(4a); then substitute u  x  b/(2a) and B  (4ac  b2)/(4a).

DIFFERENTIAL AND INTEGRAL CALCULUS

PARTIAL FRACTIONS 5. For a ratio of polynomials, where the denominator has been completely factored into linear factors pi(x) and quadratic factors qj(x), and where the degree of the numerator is less than the degree of the denominator, then rewrite r(x)/[p1(x) . . . pn(x)q1(x) . . . qm(x)]  A1 /p1(x)  . . .  An /pn(x)  (B1x  C1)/q1(x) . . .  (Bm x  Cm)/qm(x). INTEGRATION BY PARTS 6. Change the integral using the formula 3 u dv 5 uv 2 3 v du where u and dv are chosen so that (a) v is easy to find from dv, and (b) v du is easier to find than u dv. Kasube suggests (“A Technique for Integration by Parts,” Am. Math. Month., vol. 90, no. 3, Mar. 1983): Choose u in the order of preference LIATE, that is, Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. EXAMPLE. Find 3 x ln x dx. The logarithmic ln x has higher priority than does the algebraic x, so let u  ln (x) and dv  x dx. Then du  (1/x) dx; v 5 x 2/2, so 3 x ln x dx 5 uv 2 3 v du 5 sx 2/2d ln x 2 3 sx 2/2ds1/xd dx 5 (x2/2)

Properties of Definite Integrals

Definite Integrals The definite integral of f (x) dx from x  a to b

b

a

a

b

3 5 23 ;

c

b

b

a

c

a

3 1 3 5 3

MEAN-VALUE THEOREM FOR INTEGRALS b

b

a

a

3 Fsxdf sxd dx 5 FsXd 3 f sxd dx provided f(x) does not change sign from x  a to x  b; here X is some (unknown) value of x intermediate between a and b. MEAN VALUE. The mean value of f(x) with respect to x, between a and b, is f 5

b 1 f sxd dx 3 b2a a x5b

THEOREM ON CHANGE OF VARIABLE. In evaluating 3

fsxd dx, f(x) dx

x5a

may be replaced by its value in terms of a new variable t and dt, and x  a and x  b by the corresponding values of t, provided that throughout the interval the relation between x and t is a one-to-one correspondence (i.e., to each value of x there corresponds one and only one value of t, and to each value of t there corresponds one and only x5b

one value of x). So 3

t5gsbd

f sxd dx 5 3

x5a

ln x 2 3 x/2 dx 5 sx 2/2d ln x 2 x 2/4 1 C.

x  b, denoted by 3 f sxd dx, is the limit (as n increases indefinitely)

2-29

f sgstdd grstd dt.

t5gsad

DIFFERENTIATION WITH RESPECT TO THE UPPER LIMIT. If b is variable, then b

3 f sxd dx is a function b, whose derivative is a

d b f sxd dx 5 f sbd db 3a

a

of a sum of n terms: b

[ f sx1d x 1 f sx2d x 1 f sx3d x 1 # # # 1f sxnd x] 3 f sxd dx 5 nlim S`

DIFFERENTIATION WITH RESPECT TO A PARAMETER

a

built up as follows: Divide the interval from a to b into n equal parts, and call each part  x,  (b  a)/n; in each of these intervals take a value of x (say, x1, x2, . . . , xn), find the value of the function f(x) at each of these points, and multiply it by x, the width of the interval; then take the limit of the sum of the terms thus formed, when the number of terms increases indefinitely, while each individual term approaches zero. b

Geometrically, 3 fsxd dx is the area bounded by the curve y  f(x), a

the x axis, and the ordinates x  a and x  b (Fig. 2.1.108); i.e., briefly, the “area under the curve, from a to b.” The fundamental theorem for the evaluation of a definite integral is the following:

b 'f sx, cd ' b f sx, cd dx 5 3 dx 'c 3a 'c a

Functions Defined by Definite Integrals The following definite

integrals have received special names:

when k2 1.

a

i.e., the definite integral is equal to the difference between two values of any one of the indefinite integrals of the function in question. In other words, the limit of a sum can be found whenever the function can be integrated.

Fig. 2.1.108 Graph showing areas to be summed during integration.

dx 21 2 k 2 sin2 x u

2. Elliptic integral of the second kind 5 Esu, kd 5 3 21 2 k 2 sin2 x 0

dx, when k2 1. 3, 4. Complete elliptic integrals of the first and second kinds; put u  p/2 in (1) and (2).

b

3 f sxd dx 5 B 3 f sxd dxR x5b 2 B 3 f sxd dxR x5a

u

0

1. Elliptic integral of the first kind  Fsu, kd 5 3

5. The probability integral 5

2 2p

x

2x 3 e dx. 2

0 `

6. The gamma function  snd 5 3 x n21e2x dx. 0

Approximate Methods of Integration. Mechanical Quadrature

(See also section “Numerical Methods.”) 1. Use Simpson’s rule (see also Scarborough, “Numerical Mathematical Analyses,” Johns Hopkins Press). 2. Expand the function in a converging power series, and integrate term by term. 3. Plot the area under the curve y  f(x) from x  a to x  b on squared paper, and measure this area roughly by “counting squares.” Double Integrals The notation   f(x, y) dy dx means [ f(x, y) dy] dx, the limits of integration in the inner, or first, integral being functions of x (or constants).

2-30

MATHEMATICS

EXAMPLE. To find the weight of a plane area whose density, w, is variable, say w  f(x, y). The weight of a typical element, dx dy, is f(x, y) dx dy. Keeping x and dx constant and summing these elements from, say, y  F1(x) to y  F2(x), as determined by the shape of the boundary (Fig. 2.1.109), the weight of a typical strip perpendicular to the x axis is y5F2sxd

dx 3

f sx, yd dy

y5F1sxd

Finally, summing these strips from, say, x  a to x  b, the weight of the whole area is x5b

3 x5a

y5F2sxd

B dx 3

f sx, yd dy R or, briefly, 3 3 f sx, yd dy dx

y5F1sxd

Fig. 2.1.109 Graph showing areas to be summed during double integration.

xn has limit zero. A series of partial sums of an alternating sequence is called an alternating series. THEOREM. An alternating series converges whenever the sequence xn has limit zero. A series is a geometric series if its terms are of the form ar n. The value r is called the ratio of the series. Usually, for geometric series, the index is taken to start with n  0 instead of n  1. THEOREM. A geometric series with xn  arn, n  0, 1, 2, . . . , converges if and only if 1 r 1, and then the limit of the series is a/(1  r). The partial sums of a geometric series are sn  a(1  r n)/ (1  r). The series defined by the sequence xn  1/n, n  1, 2, . . . , is called the harmonic series. The harmonic series diverges. A series with each term xn  0 is called a “positive series.” There are a number of tests to determine whether or not a positive series sn converges. 1. Comparison test. If c1  c2      cn is a positive series that converges, and if 0 xn cn, then the series x1  x2      xn also converges. If d1  d2      dn diverges and xn  dn, then x1  x2      xn also diverges. 2. Integral test. If f(t) is a strictly decreasing function and f(n)  xn, `

then the series sn and the integral 3 f std dt either both converge or both 1

Triple Integrals The notation 3 3 3 f sx, y, zd dz dy dx means

3 b 3 B 3 f sx, y, zd dz R dyr dx Such integrals are known as volume integrals. EXAMPLE. To find the mass of a volume which has variable density, say, w  f(x, y, z). If the shape of the volume is described by a x b, F1(x) y F2(x), and G1(x, y) z G2(x, y), then the mass is given by F2sxd

b

3 3 a

F1sxd

G2sx, yd

3

diverge. 3. P test. The series defined by xn  1/np converges if p  1 and diverges if p  1 or p 1. If p  1, then this is the harmonic series. 4. Ratio test. If the limit of the sequence xn1/xn  r, then the series diverges if r  1, and it converges if 0 r 1. The test is inconclusive if r  1. 5. Cauchy root test. If L is the limit of the nth root of the nth term, lim x 1/n n , then the series converges if L 1 and diverges if L  1. If L  1, then the test is inconclusive. A power series is an expression of the form a0  a1x  a2x2      `

f sx, y, zd dz dy dx

G1sx, yd

SERIES AND SEQUENCES Sequences

A sequence is an ordered list of numbers, x1, x2, . . . , xn, . . . . An infinite sequence is an infinitely long list. A sequence is often defined by a function f(n), n  1, 2, . . . . The formula defining f(n) is called the general term of the sequence. The variable n is called the index of the sequence. Sometimes the index is taken to start with n  0 instead of n  1. A sequence converges to a limit L if the general term f(n) has limit L as n goes to infinity. If a sequence does not have a unique limit, the sequence is said to “diverge.” There are two fundamental ways a function can diverge: (1) It may become infinitely large, in which case the sequence is said to be “unbounded,” or (2) it may tend to alternate among two or more values, as in the sequence xn  (1)n. A sequence alternates if its odd-numbered terms are positive and its even-numbered terms are negative, or vice versa. Series

A series is a sequence of sums. The terms of the sums are another sequence, x1, x2, . . . . Then the series is the sequence defined by n

sn  x1  x2      xn  g xi. The sequence sn is also called the i51

sequence of partial sums of the series.

If the sequence of partial sums converges (resp. diverges), then the series is said to converge (resp. diverge). If the limit of a series is S, then the sequence defined by rn  S  sn is called the “error sequence” or the “sequence of truncation errors.” Convergence of Series THEOREM. If a series sn  x1  x2      xn converges, then it is necessary (but not sufficient) that the sequence

anxn     or g aix i. i50

The range of values of x for which a power series converges is the interval of convergence of the power series. General Formulas of Maclaurin and Taylor If f(x) and all its derivatives are continuous in the neighborhood of the point x  0 (or x  a), then, for any value of x in this neighborhood, the function f(x) may be expressed as a power series arranged according to ascending powers of x (or of x  a), as follows: f sxd 5 f s0d 1

f sxd 5 f sad 1

f rs0d f - s0d 3 f ss0d 2 x1 x 1 x 1c 1! 2! 3! f sn21d s0d n21 1 sPndx n 1 x sn 2 1d!

(Maclaurin)

f rsad f - sad f ssad sx 2 ad 1 sx 2 ad2 1 sx 2 ad31 1! 2! 3!

c1

f sn21d sad sn 2 1d!

sx 2 adn21 1 sQ ndsx 2 adn

(Taylor)

Here (Pn)xn, or (Qn)(x  a)n, is called the remainder term; the values of the coefficients Pn and Qn may be expressed as follows: Pn 5 [ f snd ssxd]/n! 5 [s1 2 tdn21f snd stxd]/sn 2 1d!

Q n 5 5 f snd[a 1 ssx 2 ad]6/n!

5 5 s1 2 tdn21f snd[a 1 tsx 2 ad]6/sn 2 1d!

where s and t are certain unknown numbers between 0 and 1; the s form is due to Lagrange, the t form to Cauchy. The error due to neglecting the remainder term is less than sPndx n, or sQ nd(x  a)n, where Pn, or Q n, is the largest value taken on by Pn, or

ORDINARY DIFFERENTIAL EQUATIONS

Qn, when s or t ranges from 0 to 1. If this error, which depends on both n and x, approaches 0 as n increases (for any given value of x), then the general expression with remainder becomes (for that value of x) a convergent infinite series. The sum of the first few terms of Maclaurin’s series gives a good approximation to f(x) for values of x near x  0; Taylor’s series gives a similar approximation for values near x  a. The MacLaurin series of some important functions are given below. Power series may be differentiated term by term, so the derivative of a power series a0  a1x  a2x2      an x n is a1  2a2x      naxxn  1 . . . . The power series of the derivative has the same interval of convergence, except that the endpoints may or may not be included in the interval.

`

1 5 g xn 12x n50 ` sm 1 n 2 1d! 1 511 g xn m s1 2 xd n51 sm 2 1d!n!

ln a

x x2 x3 x4 1 1 1 1c 1! 2! 3! 4! 2

tan 21 y 5 y 2

y3 y5 y7 1 2 1c 7 3 5

[21 # y # 11]

cot 21 y 5 1⁄2p 2 tan21 y.

x4 x6 x2 1c 1 1 2! 4! 6!

[2` , x , `]

21 , x , 1

sinh21 y 5 y 2

y3 3y 5 5y 7 1 2 1c 40 112 6

[21 , y , 11]

tanh21 y 5 y 1

y3 y7 y5 1c 1 1 7 3 5

[21 , y , 11]

3

x3 x4 x5 c x2 1 2 1 2 3 4 5

[21 , x , 11]

x3 x4 x5 c x2 2 2 2 2 2 3 4 5

[21 , x , 11] [21 , x , 11]

x11 1 1 1 1 b 5 2a x 1 3 1 5 1 7 1 c b x21 7x 3x 5x [x , 21 or 11 , x] x21 1 x21 1 x21 1 a b 1 a b 1 cd x11 3 x11 5 x11 3

5

[0 , x , `] 3 x x 1 ln sa 1 xd 5 ln a 1 2 c 1 a b 2a 1 x 3 2a 1 x 5 x 1 a b 1 cd 5 2a 1 x

Series for the Trigonometric Functions In the following formulas, all angles must be expressed in radians. If D  the number of degrees in the angle, and x  its radian measure, then x  0.017453D.

x3 x5 x7 1 2 1c 3! 7! 5! x2 x4 x6 x8 cos x 5 1 2 1 2 1 2c 2! 4! 8! 6!

ORDINARY DIFFERENTIAL EQUATIONS

An ordinary differential equation is one which contains a single independent variable, or argument, and a single dependent variable, or function, with its derivatives of various orders. A partial differential equation is one which contains a function of several independent variables, and its partial derivatives of various orders. The order of a differential equation is the order of the highest derivative which occurs in it. A solution of a differential equation is any relation among the variables, involving no derivatives, though possibly involving integrations which, when substituted in the given equation, will satisfy it. The general solution of an ordinary differential equation of the nth order will contain n arbitrary constants. If specific values of the arbitrary constants are chosen, then a solution is called a particular solution. For most problems, all possible particular solutions to a differential equation may be found by choosing values for the constants in a general solution. In some cases, however, other solutions exist. These are called singular solutions. EXAMPLE. The differential equation (yy)2  a2  y2  0 has general solution (x  c)2  y2  a2, where c is an arbitrary constant. Additionally, it has the two singular solutions y  a and y  a. The singular solutions form two parallel lines tangent to the family of circles given by the general solution.

The example illustrates a general property of singular solutions; at each point on a singular solution, the singular solution is tangent to some curve given in the general solution. Methods of Solving Ordinary Differential Equations

[0 , a , 1`, 2a , x , 1`]

sin x 5 x 2

[21 # y # 11]

cosh x 5 1 1

m m 3 c m 2 x1 x 1 x 1 1! 2! 3! [a . 0, 2` , x , 1` ]

1

y3 5y 7 3y 5 1 1c 1 40 112 6

21 , x , 1

[2` , x , 1` ]

11x x7 x3 x5 1 1 cb b 5 2ax 1 1 7 12x 3 5

ln x 5 2 c

sin 21 y 5 y 1

[2` , x , `]

where m  ln a  (2.3026)(log10 a).

ln a

[2p , x , 1p]

x3 x5 x7 1 1 1c 3! 7! 5!

Exponential and Logarithmic Series

ln s1 2 xd 5 2x 2

2x 5 x7 x3 x 1 2 2 2c cot x 5 x 2 2 3 45 945 4725

sinh x 5 x 1

Geometrical Series

ln s1 1 xd 5 x 2

[2p/2 , x , 1p/2]

Series for the Hyperbolic Functions (x a pure number)

The range of values of x for which each of the series is convergent is stated at the right of the series.

a x 5 emx 5 1 1

62x 9 x3 2x 5 17x 7 1 1 1 1c 3 15 315 2835

cos21 y 5 1⁄2p 2 sin21 y;

Series Expansions of Some Important Functions

ex 5 1 1

tan x 5 x 1

2-31

[2` , x , 1`] [2` , x , 1`]

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 1. If possible, separate the variables; i.e., collect all the x’s and dx on one side, and all the y’s and dy on the other side; then integrate both sides, and add the constant of integration. 2. If the equation is homogeneous in x and y, the value of dy/dx in terms of x and y will be of the form dy/dx  f(y/x). Substituting y  xt will enable the variables to be separated. dt Solution: log e x 5 3 1 C. f std 2 t

2-32

MATHEMATICS

3. The expression f(x, y) dx  F(x, y) dy is an exact differential if 'fsx, yd 'Fsx, yd ( P, say). In this case the solution of f(x, y) dx  5 'y 'x F(x, y) dy  0 is f sx, yd dx 1 [Fsx, yd 2 P dx] dy 5 C Fsx, yd dy 1 [ f sx, yd 2 P dy] dx 5 C

or

dx 4. Linear differential equation of the first order: 1 f sxd # y 5 dx F(x). Solution: y 5 e2P[ePFsxd dx 1 C ], where P 5 f sxd dx dy 1 f sxd # y 5 Fsxd # y n. Substituting dx y1  n  v gives (dv/dx)  (1  n)f(x) . v  (1  n)F(x), which is linear in v and x. 6. Clairaut’s equation: y  xp  f (p), where p  dy/dx. The solution consists of the family of lines given by y  Cx  f(C), where C is any constant, together with the curve obtained by eliminating p between the equations y  xp  f( p) and x  f ( p)  0, where f ( p) is the derivative of f ( p). 7. Riccati’s equation. p  ay2  Q(x)y  R(x)  0, where p  dy/dx can be reduced to a second-order linear differential equation (d2u/dx2)  Q(x)(du/dx)  R(x)  0 by the substitution y  du/dx. 8. Homogeneous equations. A function f (x, y) is homogeneous of degree n if f(rx, ry)  rm f(x, y), for all values of r, x, and y. In practice, this means that f(x, y) looks like a polynomial in the two variables x and y, and each term of the polynomial has total degree m. A differential equation is homogeneous if it has the form f(x, y)  0, with f homogeneous. (xy  x2) dx  y2 dy  0 is homogeneous. Cos (xy) dx  y2 dy  0 is not. If an equation is homogeneous, then either of the substitutions y  vx or x  vy will transform the equation into a separable equation. 9. dy/dx  f [(ax  by  c)/(dx  ey  g)] is reduced to a homogeneous equation by substituting u  ax  by  c, v  dx  ey  g, if ae  bd  0, and z  ax  by, w  dx  ey if ae  bd  0. 5. Bernoulli’s equation:

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 10. Dependent variable missing. If an equation does not involve the variable y, and is of the form F(x, dy/dx, d 2y/dx2)  0, then it can be reduced to a first-order equation by substituting p  dy/dx and dp/dx  d 2y/dx2. 11. Independent variable missing. If the equation is of the form F(y, dy/dx, d 2y/dx 2)  0, and so is missing the variable x, then it can be reduced to a first-order equation by substituting p  dy/dx and p(dp/dy)  d 2y/dx2. d 2y 5 2n2y. 12. dx 2 Solution: y  C1 sin (nx  C2), or y  C3 sin nx  C4 cos nx. d 2y 5 1n2y. 13. dx 2 Solution: y  C1 sinh (nx  C2), or y  C3 enx  C4enx. d 2y 5 f syd. 14. dx 2 dy 1 C2, where P 5 3 f syd dy. Solution: x 5 3 2C1 1 2P d 2y 5 fsxd. 15. dx 2 Solution: or

y 5 3 P dx 1 C1x 1 C2

where P 5 3 f sxd dx,

y 5 xP 2 3 x f sxd dx 1 C1x 1 C2.

dy dy d 2y dz ≤ . Putting 5 z, 2 5 , dx dx dx dx dz zdz x5 3 and y5 3 1 C1, 1 C2 fszd fszd then eliminate z from these two equations. dy d 2y 1 a 2y 5 0. 17. The equation for damped vibrations: 2 1 2b dx dx CASE 1. If a2  b2  0, let m 5 2a 2 2 b 2. Solution: 16.

d 2y

dx 2

5f¢

y 5 C1e2bx sin smx 1 C2d or

y 5 e2bx[C3 sin smxd 1 C4 cos smxd]

CASE 2. If a2  b2  0, solution is y  ebx(C1  C2 x). CASE 3. If a2  b2 0, let n 5 2b 2 2 a 2. Solution: y 5 C1e2bx sinh snx 1 C2d or y 5 C3e2sb1ndx 1 C4e2sb2ndx dy d 2y 1 2b 1 a 2y 5 c. 18. dx dx 2 c Solution: y 5 2 1 y1, where y1  the solution of the corresponding a equation with second member zero [see type 17 above]. dy d 2y 1 2b 1 a 2y 5 c sin skxd. 19. dx dx 2 Solution: y  R sin (kx  S)  y1 where R 5 c/ 2sa 2 2 k 2d2 1 4b 2k 2, tan S  2bk/(a2  k2), and y1  the solution of the corresponding equation with second member zero [see type 17 above]. dy d 2y 1 2b 1 a 2y 5 fsxd. 20. dx dx 2 Solution: y  R sin (kx  S)  y1 where R 5 c/ 2sa 2 2 k 2d2 1 4b 2k 2, tan S  2bk/(a2  k2), and y1  the solution of the corresponding equation with second member zero [see type 17 above]. If b2 a2, 1 B em1x 3 e2m1x f sxd dx 2 em2x 3 e2m2x f sxd dxR 2 2b 2 2 a 2 where m1 5 2b 1 2b 2 2 a 2 and m2 5 2b 2 2b 2 2 a 2. y0 5

If b2 a2, let m 5 2a 2 2 b 2, then 1 y0 5 m e2bx B sin smxd 3 ebx cos smxd # f sxd dx 2 cos smxd 3 ebx sin smxd # f sxd dxd If b 2 5 a 2, y0 5 e2bx B x 3 ebxf sxd dx 2 3 x # ebx f sxd dxR. Types 17 to 20 are examples of linear differential equations with constant coefficients. The solutions of such equations are often found most simply by the use of Laplace transforms. (See Franklin, “Fourier Methods,” pp. 198–229, McGraw-Hill or Kreyszig, “Advanced Engineering Mathematics,” Wiley.) Linear Equations

For the linear equation of the nth order An sxd d ny/dx n 1 An21 sxd d n21y/dx n21 1 c 1 A1 sxddy/dx 1 A0 sxdy 5 Esxd the general solution is y  u  c1u1  c2u2      cnun. Here u, the particular integral, is any solution of the given equation, and u1, u2, . . . , un form a fundamental system of solutions of the homogeneous equation obtained by replacing E(x) by zero. A set of solutions is fundamental, or independent, if its Wronskian determinant W(x) is not

ORDINARY DIFFERENTIAL EQUATIONS

zero, where u1 u r1 # Wsxd 5 6 # #

u2 u r2 # # #

u sn21d u sn21d 1 2

c u n c ur n c # c # 6 c # c u sn21d n

For any n functions, W(x)  0 if some one ui is linearly dependent on the others, as un  k1u1  k2u2      kn1un1 with the coefficients ki constant. And for n solutions of a linear differential equation of the nth order, if W(x)  0, the solutions are linearly independent. Constant Coefficients To solve the homogeneous equation of the nth order Andny/dxn  An1 dn1 y/dxn1      A1dy/dx  A0 y  0, An  0, where An, An1, . . . , A0 are constants, find the roots of the auxiliary equation An p n 1 An21 p n21 1 c1 A1 p 1 A0 5 0 For each simple real root r, there is a term cerx in the solution. The terms of the solution are to be added together. When r occurs twice among the n roots of the auxiliary equation, the corresponding term is erx(c1  c2x). When r occurs three times, the corresponding term is erx(c1  c2x  c3x2), and so forth. When there is a pair of conjugate complex roots a  bi and a  bi, the real form of the terms in the solution is eax(c1 cos bx  d1 sin bx). When the same pair occurs twice, the corresponding term is eax[(c1  c2 x) cos bx  (d1  d2 x) sin bx], and so forth. Consider next the general nonhomogeneous linear differential equation of order n, with constant coefficients, or And ny/dx n 1 An21d n21y/dx n21 1 # # # 1 A1dy/dx 1 A0 y 5 Esxd We may solve this by adding any particular integral to the complementary function, or general solution, of the homogeneous equation obtained by replacing E(x) by zero. The complementary function may be found from the rules just given. And the particular integral may be found by the methods of the following paragraphs. Undetermined Coefficients In the last equation, let the right member E(x) be a sum of terms each of which is of the type k, k cos bx, k sin bx, keax, kx, or more generally, kxmeax, kxmeax cos bx, or kxmeax sin bx. Here m is zero or a positive integer, and a and b are any real numbers. Then the form of the particular integral I may be predicted by the following rules. CASE 1. E(x) is a single term T. Let D be written for d/dx, so that the given equation is P(D)y  E(x), where P(D)  AnDn  An  1Dn  1      A1D  A0 y. With the term T associate the simplest polynomial Q(D) such that Q(D)T  0. For the particular types k, etc., Q(D) will be D, D2  b2, D2  b2, D  a, D2; and for the general types kxm eax, etc., Q(D) will be (D  a)m1, (D2  2aD  a2  b2)m+1, (D2  2aD  a2  b2)m1. Thus Q(D) will always be some power of a first- or second-degree factor, Q(D)  Fv, F  D  a, or F  D2 – 2aD  a2  b2. Use the method described under Constant Coefficients to find the terms in the solution of P(D)y  0 and also the terms in the solution of Q(D)P(D)y  0. Then assume the particular integral I is a linear combination with unknown coefficients of those terms in the solution of Q(D)P(D)y  0 which are not in the solution of P(D)y  0. Thus if Q(D)  Fq and F is not a factor of P(D), assume I  (Ax q1  Bx q2      L)eax when F  D  a, and assume I  (Ax q1  Bx q2      L)eax cos bx  (Mx q1  Nx q2      R)eax sin bx when F  D2  2aD  a2  b2. When F is a factor of P(D) and the highest power of F which is a divisor of P(D) is Fk, try the I above multiplied by xk. CASE 2. E(x) is a sum of terms. With each term in E(x), associate a polynomial Q(D)  Fq as before. Arrange in one group all the terms that have the same F. The particular integral of the given equation will be the sum of solutions of equations each of which has one group on the right. For any one such equation, the form of the particular integral

2-33

is given as for Case 1, with q the highest power of F associated with any term of the group on the right. After the form has been found in Case 1 or 2, the unknown coefficients follow when we substitute back in the given differential equation, equate coefficients of like terms, and solve the resulting system of simultaneous equations. Variation of Parameters. Whenever a fundamental system of solutions u1, u2, . . . , un for the homogeneous equation is known, a particular integral of An sxdd ny/dx n 1 An21 sxdd n21y/dx n21 1 c 1 A1 sxddy/dx 1 A0 sxdy 5 Esxd may be found in the form y 5 vkuk. In this and the next few summations, k runs from 1 to n. The vk are functions of x, found by integrating their derivatives vrk, and these derivatives are the solutions of the n simultaneous equations v kr uk 5 0, v kr u kr 5 0, v kr us k 5 0, c , v rku sn22d 5 0, An sxdv rku sn21d 5 Esxd. To find the vk from vk  k k v kr dx 1 ck, any choice of constants will lead to a particular integral. x

The special choice vk 5 3 v rk dx leads to the particular integral having 0

y, y, y, . . . . , y (n 1) each equal to zero when x  0. The Cauchy-Euler Equidimensional Equation This has the form knx nd ny/dx n 1 kn21x n21d n21y/dx n21 1 c 1 k1 x dy/dx 1 k0 y 5 Fsxd

The substitution x  et, which makes x dy/dx 5 dy/dt x kd ky/dx k 5 sd/dt 2 k 1 1d c sd/dt 2 2dsd/dt 2 1d dy/dt transforms this into a linear differential equation with constant coefficients. Its solution y  g(t) leads to y  g(ln x) as the solution of the given Cauchy-Euler equation. Bessel’s Equation The general Bessel equation of order n is: x 2ys 1 xyr 1 sx 2 2 n2dy 5 0 This equation has general solution y 5 AJn sxd 1 BJ2n sxd when n is not an integer. Here, Jn(x) and Jn(x) are Bessel functions (see section on Special Functions). In case n  0, Bessel’s equation has solution ` s21dkH x 2k k y 5 AJ0 sxd 1 B B J0 sxd ln sxd 2 g R 22k sk!d2 k51 where Hk is the kth partial sum of the harmonic series, 1  1⁄2  1⁄3      1/k. In case n  1, the solution is y 5 AJ1 sxd 1 B bJ1 sxd ln sxd 1 1/x `

s21dk sHk 1 Hk21dx 2k21

k51

22kk!sk 2 1d!

2 Bg

Rr

In case n  1, n is an integer, solution is `

s21dk11 sn 2 1d!x 2k2n R 2k112n k!s1 2 ndk k50 2 2k1n ` s21dk11 sH 1 H k k11dx 1 1⁄2 B g Rr 2k1n 2 k!sk 1 nd! k50

y 5 AJn sxd 1 B bJn sxd ln sxd 1 B g

Solutions to Bessel’s equation may be given in several other forms, often exploiting the relation between Hk and ln (k) or the so-called Euler constant. General Method of Power Series Given a general differential equation F(x, y, y, . . .)  0, the solution may be expanded as a Maclaurin series, so y 5 `n50anx n, where an  f (n)(0)/n!. The power

2-34

MATHEMATICS

series for y may be differentiated formally, so that yr 5 `n51nanx n21 5 `n50 sn 1 1d an11x n, and ys 5 `n52 nsn 2 1d anx n22 5 `n50 sn 1 1d sn 1 2dan12 x n. Substituting these series into the equation F(x, y, y, . . .)  0 often gives useful recursive relationships, giving the value of an in terms of previous values. If approximate solutions are useful, then it may be sufficient to take the first few terms of the Maclaurin series as a solution to the equation. EXAMPLE. Consider y  y  xy  0. The procedure gives `n50 sn 1 1d sn 1 2d an12 x n 2 `n50 sn 1 1d an11 x n 1 x `n50 an x n 5 `n50 sn 1 1d sn 1 2dan12 x n 2 `n50 sn 1 1d an11 x n 1 `n51 an21 x n 5 s2a2 2 a1dx 0 1 `n51 [sn 1 1dsn 1 2dan12 2 sn 1 1dan11 1 an21] x n 5 0. Thus 2a2  a1  0 and, for n  0, (n  1)(n  2)an2  (n  1) an1  an1  0. Thus, a0 and a1 may be determined arbitrarily, but thereafter, the values of an are determined recursively.

PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations (PDEs) arise when there are two or more independent variables. Two notations are common for the partial derivatives involved in PDEs, the “del” or fraction notation, where the first partial derivative of f with respect to x would be written ' f/'x, and the subscript notation, where it would be written fx. In the same way that ordinary differential equations often involve arbitrary constants, solutions to PDEs often involve arbitrary functions. EXAMPLE. fxy  0 has as its general solution g(x)  h(y). The function g does not depend on y, so gy  0. Similarly, fx  0.

PDEs usually involve boundary or initial conditions dictated by the application. These are analogous to initial conditions in ordinary differential equations. In solving PDEs, it is seldom feasible to find a general solution and then specialize that general solution to satisfy the boundary conditions, as is done with ordinary differential equations. Instead, the boundary conditions usually play a key role in the solution of a problem. A notable exception to this is the case of linear, homogeneous PDEs since they have the property that if f1 and f2 are solutions, then f1  f2 is also a solution. The wave equation is one such equation, and this property is the key to the solution described in the section “Fourier Series.” Often it is difficult to find exact solutions to PDEs, so it is necessary to resort to approximations or numerical solutions. (See also “Numerical Methods” below.) Classification of PDEs Linear A PDE is linear if it involves only first derivatives, and then only to the first power. The general form of a linear PDE, in two independent variables, x and y, and the dependent variable z, is P(x, y, z) fx  Q(x, y, z)fy  R(x, y, z), and it will have a solution of the form z  f (x, y) if its solution is a function, or F(x, y, z)  0 if the solution is not a function. Elliptic Laplace’s equation fxx  fyy  0 and Poisson’s equation fxx  fyy  g(x, y) are the prototypical elliptic equations. They have analogs in more than two variables. They do not explicitly involve the variable time and generally describe steady-state or equilibrium conditions, gravitational potential, where boundary conditions are distributions of mass, electrical potential, where boundary conditions are electrical charges, or equilibrium temperatures, and where boundary conditions are points where the temperature is held constant. Parabolic Tt  Txx  Tyy represents the dynamic condition of diffusion or heat conduction, where T(x, y, t) usually represents the temperature at time t at the point (x, y). Note that when the system reaches steady state, the temperature is no longer changing, so Tt  0, and this becomes Laplace’s equation. Hyperbolic Wave propagation is described by equations of the type utt  c2(uxx  uyy), where c is the velocity of waves in the medium.

VECTOR CALCULUS Vector Fields A vector field is a function that assigns a vector to each point in a region. If the region is two-dimensional, then the vectors assigned are two-dimensional, and the vector field is a two-dimensional vector field, denoted F(x, y). In the same way, a three-dimensional vector field is denoted F(x, y, z). A three-dimensional vector field can always be written:

Fsx, y, zd 5 f1 sx, y, zdi 1 f2 sx, y, zdj 1 f3 sx, y, zdk where i, j, and k are the basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively. The functions f1, f2, and f3 are called coordinate functions of F. Gradient If f(x, y) is a function of two variables, then the gradient is the associated vector field grad s f d 5 fxi 1 fy j. Similarly, the gradient of a function of three variables is fxi 1 fy j 1 fzk. At any point, the direction of the gradient vector points in the direction in which the function is increasing most rapidly, and the magnitude is the rate of increase in that direction. Parameterized Curves If C is a curve from a point A to a point B, either in two dimensions or in three dimensions, then a parameterization of C is a vector-valued function r(t)  r1(t)i  r2(t)j  r3(t)k, which satisfies r(a)  A, r(b)  B, and r(t) is on the curve C, for a # t # b. It is also necessary that the function r(t) be continuous and one-to-one. A given curve C has many different parameterizations. The derivative of a parameterization r(t) is a vector-valued function r r std 5 r r1 stdi 1 r r2 stdj 1 r r3 stdk. The derivative is the velocity function of the parameterization. It is always tangent to the curve C, and the magnitude is the speed of the parameterization. Line Integrals If F is a vector field, C is a curve, and r(t) is a parameterization of C, then the line integral, or work integral, of F along C is b

W 5 3 F ? dr 5 3 Fsrstdd ? rrstd dt a

C

This is sometimes called the work integral because if F is a force field, then W is the amount of work necessary to move an object along the curve C from A to B. Divergence and Curl The divergence of a vector field F is div F  f1x  f2y  f3z. If F represents the flow of a fluid, then the divergence at a point represents the rate at which the fluid is expanding at that point. Vector fields with div F  0 are called incompressible. The curl of F is curl F 5 s f3y 2 f2z d i 1 s f1z 2 f3x d j 1 s f2x 2 f1y dk If F is a two-dimensional vector field, then the first two terms of the curl are zero, so the curl is just curl F 5 s f2x 2 f1y d k If F represents the flow of a fluid, then the curl represents the rotation of the fluid at a given point. Vector fields with curl F  0 are called irrotational.

Two important facts relate div, grad, and curl: 1. div (curl F)  0 2. curl (grad f )  0 Conservative Vector Fields A vector field F  f1i  f2 j  f3k is conservative if all of the following are satisfied: f1y 5 f2x

f1z 5 f3x

and

f2z 5 f3y

If F is a two-dimensional vector field, then the second and third conditions are always satisfied, and so only the first condition must be checked. Conservative vector fields have three important properties: 1. 3 F # dr has the same value regardless of what curve C is chosen that C

connects the points A and B. This property is called path independence. 2. F is the gradient of some function f (x, y, z). 3. Curl F  0.

LAPLACE AND FOURIER TRANSFORMS

In the special case that F is a conservative vector field, if F  grad ( f ), then # 3 F dr 5 f sBd 2 f sAd

Properties of Laplace Transforms Fssd 5 ls fstdd

f(t)

THEOREMS ABOUT LINE AND SURFACE INTEGRALS

Two important theorems relate line integrals with double integrals. If R is a region in the plane and if C is the curve tracing the boundary of R in the positive (counterclockwise) direction, and if F is a continuous vector field with continuous first partial derivatives, line integrals on C are related to double integrals on R by Green’s theorem and the divergence theorem. Green’s Theorem

# # 3 F dr 5 3 3 curl sFd dS R

The right-hand double integral may also be written as 3 3 |curl sFd| dA.

2. 3. 4. 5.

Name of rule

`

2st 3 e f std dt

1. f(t)

C

C

Table 2.1.5

2-35

Definition

0

fstd 1 gstd k f (t) f rstd f sstd

6. f -std 7. 3f std dt

Fssd 1 Gssd kF(s) sFssd 2 f s01d s 2Fssd 2 sf s01d 2 f rs01d s 3Fssd 2 s 2f s01d 2 sf rs01d 2 f ss01d

Addition Scalar multiples Derivative laws

s1/sdFssd

Integral law

1 s1/sd 3f std dt|01 8. 9. 10. 11. 12.

f(bt) eatf (t) f * gstd ua stdfst 2 ad 2 t f st d

(1/b)F(s/b) Fss 2 ad F(s)G(s) Fssde2at Frssd

Change of scale First shifting Convolution Second shifting Derivative in s

R

Green’s theorem describes the total rotation of a vector field in two different ways, on the left in terms of the boundary of the region and on the right in terms of the rotation at each point within the region. Divergence Theorem

# 3 F d N 5 3 3 div sFd dA C

R

where N is the so-called normal vector field to the curve C. The divergence theorem describes the expansion of a region in two distinct ways, on the left in terms of the flux across the boundary of the region and on the right in terms of the expansion at each point within the region. Both Green’s theorem and the divergence theorem have corresponding theorems involving surface integrals and volume integrals in three dimensions. LAPLACE AND FOURIER TRANSFORMS Laplace Transforms The Laplace transform is used to convert equations involving a time variable t into equations involving a frequency

Table 2.1.4

Fssd 5 ls fstdd

12. 13. 14. 15.

a/s 1/s 0 t , a e2as/S ua std 5 e 1 t.a da std 5 u ra std e2as at e 1/ss 2 ad s1>rde2t/r 1/srs 1 1d 2at ke k/ss 1 ad a/ss 2 1 a 2d sin at s/ss 2 1 a 2d cos at b/[ss 1 ad2 1 b 2] e2at sin bt e at ebt 1 2 a2b a2b ss 2 adss 2 bd 1/s 2 t t2 2/s 3 tn n!/s n11 ta sa 1 1d/s a11

16. 17. 18. 19. 20. 21. 22.

sinh at cosh at t neat t cos at t sin at sin at 2 at cos at arctan a/s

11.

Therefore, Fssd 5 l[ f(t)]. The Laplace integral is defined as

a/ss 2 2 a 2d s/ss 2 2 a 2d n!/ss 2 adn11 ss 2 2 a 2d/ss 2 1 a 2d2 2as/ss 2 1 a 2d2 2a 3/ss 2 1 a 2d2 ssin atd/t

`

`

0

0

l 5 3 e2st dt. Therefore, l[ f std] 5 3 e2stf std dt Name of function

1. a 2. 1

4. 5. 6. 7. 8. 9. 10.

f(t)  a function of time s  a complex variable of the form ss 1 jvd F(s)  an equation expressed in the transform variable s, resulting from operating on a function of time with the Laplace integral l  an operational symbol indicating that the quantity which it prefixes is to be transformed into the frequency domain f(0)  the limit from the positive direction of f(t) as t approaches zero  f(0 )  the limit from the negative direction of f(t) as t approaches zero

Laplace Transforms

f(t)

3.

variable s. There are essentially three reasons for doing this: (1) higherorder differential equations may be converted to purely algebraic equations, which are more easily solved; (2) boundary conditions are easily handled; and (3) the method is well-suited to the theory associated with the Nyquist stability criteria. In Laplace-transformation mathematics the following symbols and equations are used (Tables 2.1.4 and 2.1.5):

Direct Transforms EXAMPLE. f std 5 sin bt

Heavyside or step function

`

Dirac or impulse function

l[ f std] 5 ls sin btd 5 3 sin bt e2st dt 0

but

sin bt 5 l ssin btd 5 5

Gamma function (see “Special Functions”)

ejbt 2 e2jbt 2j

where

j 2 5 21

1 ` jbt se 2 e2jbtde2st dt 2j 30 ` 1 21 ¢ ≤ es2s1jbdt 2 2j s 2 jb 0

2 5

` 1 21 ¢ ≤ es2s2jbdt 2 2j s 1 jb 0

b s 2 1 b2

Table 2.1.4 lists the transforms of common time-variable expressions. Some special functions are frequently encountered when using Laplace methods.

2-36

MATHEMATICS

The Heavyside, or step, function ua(t) sometimes written u(t  a), is zero for all t a and 1 for all t  a. Its value at t  a is defined differently in different applications, as 0, 1⁄2, or 1, or it is simply left undefined. In Laplace applications, the value of a function at a single point does not matter. The Heavyside function describes a force which is “off” until time t  a and then instantly goes “on.” The Dirac delta function, or impulse function, da(t), sometimes written d(t  a), is the derivative of the Heavyside function. Its value is always zero, except at t  a, where its value is “positive infinity.” It is sometimes described as a “point mass function.” The delta function describes an impulse or an instantaneous transfer of momentum. The derivative of the Dirac delta function is called the dipole function. It is less frequently encountered. The convolution f * g(t) of two functions f (t) and g(t) is defined as t

f * gstd 5 3 f sudgst 2 ud du

output quantity or the error. The solution gained from the transformed equation is expressed in terms of the complex variable s. For many design or analysis purposes, the solution in s is sufficient, but in some cases it is necessary to retransform the solution in terms of time. The process of passing from the complex-variable (frequency domain) expression to that of time (time domain) is called an inverse transformation. It is represented symbolically as l21Fssd 5 fstd For any f(t) there is only one direct transform, F(s). For any given F(s) there is only one inverse transform f(t). Therefore, tables are generally used for determining inverse transforms. Very complete tables of inverse transforms may be found in Gardner and Barnes, “Transients in Linear Systems.” As an example of the inverse procedure consider an equation of the form xstd dt K 5 axstd 1 3 b

0

Laplace transforms are often used to solve differential equations arising from so-called linear systems. Many vibrating systems and electrical circuits are linear systems. If an input function fi(t) describes the forces exerted upon a system and a response or output function fo(t) describes the motion of the system, then the transfer function T(s)  Fo (s)/Fi(s). Linear systems have the special property that the transfer function is independent of the input function, within the elastic limits of the system. Therefore, Fo ssd Go ssd 5 Fi ssd Gi ssd This gives a technique for describing the response of a system to a complicated input function if its response to a simple input function is known. EXAMPLE. Solve y  2y  3y  8et subject to initial conditions y(0)  2 and y(0)  0. Let y  f (t) and Y  F(s). Take Laplace transforms of both sides and substitute for y(0) and y(0), and get s 2Y 2 2s 1 2ssY 2 2d 2 3Y 5

8 s21

It is desired to obtain an expression for x(t) resulting from an instantaneous change in the quantity K. Transforming the last equation yields f 21 s01d Xssd K s 5 Xssda 1 sb 1 s If f 1(0)/s  0 Xssd 5

then

xstd 5 l 21[Xssd] 5 l21

Fourier Coefficients Fourier coefficients are used to analyze periodic functions in terms of sines and cosines. If f(x) is a function with period 2L, then the Fourier coefficients are defined as

Y5

5

FoGi Fi 4ss 2 1 1d

ss 2 1 4d2 2s2d 16 12 2 R 5 2 B 16 ss 2 1 22d2 s 1 22

nps 1 L f ssd cos ds L 32L L

n 5 0, 1, 2, c

bn 5

nps 1 L f ssd sin ds L 32L L

n 5 1, 2, c

fsxd 5

y 5 e23t 1 et 1 2tet

Go ssd 5

an 5

Equivalently, the bounds of integration may be taken as 0 to 2L instead of –L to L. Many treatments take L 5 p or L 5 1. Then the Fourier theorem states that

Using the tables of transforms to find what function has Y as its transform, we get

EXAMPLE. A vibrating system responds to an input function fi(t)  sin t with a response fo (t)  sin 2t. Find the system response to the input gi(t)  sin 2t. Apply the invariance of the transfer function, and get

` a0 npx npx 1 g an cos ¢ ≤ 1 bn sin ¢ ≤ L L 2 n51

The series on the right is called the “Fourier series of the function f(x).” The convergence of the Fourier series is usually rapid, so that the function f(x) is usually well-approximated by the sum of the first few sums of the series. Examples of the Fourier Series If y  f(x) is the curve in Figs. 2.1.110 to 2.1.112, then in Fig. 2.1.110, y5

3px 5px px h 4h 1 1 cos c 1 c≤ 2 2 ¢cos c 1 cos c 1 2 9 25 p

Applying formulas 8 and 21 from Table 2.1.4 of Laplace transforms, go std 5 2 sin 2t 2 3⁄4 sin 2t 1 3⁄2 t cos 2t Inversion When an equation has been transformed, an explicit solution for the unknown may be directly determined through algebraic manipulation. In automatic-control design, the equation is usually the differential equation describing the system, and the unknown is either the

K/a s 1 1/ab

K From Table 2.1.4, xstd 5 a e2t/ab

Solve for Y, apply partial fractions, and get 2s 2 1 2s 1 4 ss 1 3dss 2 1d2 s11 1 5 1 s13 ss 2 1d2 1 1 2 5 1 1 ss 1 3d ss 2 1d ss 2 1d2

K/a s 1 1/ab

Fig. 2.1.110 Saw-tooth curve.

SPECIAL FUNCTIONS

In Fig. 2.1.111,

2-37

to describe its Fourier transform:

3px 5px px 4h 1 1 y 5 p ¢sin c 1 sin c 1 sin c 1 # # # ≤ 3 5

Aswd 5 3

`

f sxd cos wx dx

2`

Bswd 5 3

`

f sxd sin wx dx

2`

Then the Fourier integral equation is `

f sxd 5 3 Aswd cos wx 1 Bswd sin wx dw 0

The complex Fourier transform of f(x) is defined as Fig. 2.1.111 Step-function curve or square wave.

Fswd 5 3

`

f sxd eiwx dx

2`

Then the complex Fourier integral equation is

In Fig. 2.1.112, 3px px 2px 2h 1 1 y 5 p ¢ sin c 2 sin c 1 sin c 2 # # # ≤ 2 3

Fig. 2.1.112 Linear-sweep curve.

f sxd 5

Heat Equation The Fourier transform may be used to solve the onedimensional heat equation ut(x, t)  uxx(x, t), for t  0, given initial condition u(x, 0)  f(x). Let F(s) be the complex Fourier transform of f(x), and let U(s, t) be the complex Fourier transform of u(x, t). Then the transform of ut(x, t) is dU(s, t)/dt. Transforming ut(x, t)  uxx (s, t) yields dU/dt  s2U  0 and U(s, 0)  f (s). Solving this using the Laplace transform gives U(s, t)  2 F(s)es t. Applying the complex Fourier integral equation, which gives u(x, t) in terms of U(s, t), gives

usx, td 5 If the Fourier coefficients of a function f(x) are known, then the coefficients of the derivative f(x) can be found, when they exist, as follows: a rn 5 nbn

b rn 5 2nan

where a rn and b rn are the Fourier coefficients of f(x). The complex Fourier coefficients are defined by: cn 5 1⁄2 san 2 ibnd c0 5 1⁄2a0 cn 5 1⁄2 san 1 ibnd `

f sxd 5 g cnein px/L n52`

Wave Equation Fourier series are often used in the solution of the wave equation a2uxx  utt where 0 x L, t  0, and initial conditions are u(x, 0)  f(x) and ut(x, 0)  g(x). This describes the position of a vibrating string of length L, fixed at both ends, with initial position f (x) and initial velocity g(x). The constant a is the velocity at which waves are propagated along the string, and is given by a2  T/p, where T is the tension in the string and p is the mass per unit length of the string. If f(x) is extended to the interval L x L by setting f(x)   f(x), then f may be considered periodic of period 2L. Its Fourier coefficients are L npx bn 5 3 fsxd sin p dx L 2L

usx, td 5 g bn sin n51

1 ` ` 2 f sydeissy2xdes t ds dy 2p 32` 30

Applying the Euler formula, eix  cos x  i sin x, usx, td 5

1 ` ` 2 f syd cos sssy 2 xddes t ds dy 2p 32` 30

Gamma Function The gamma function is a generalization of the factorial function. It arises in Laplace transforms of polynomials, in continuous probability, applications involving fractals and fractional derivatives, and in the solution to certain differential equations. It is defined by the improper integral: `

sxd 5 3 t x21e2t dt 0

The integral converges for x  0 and diverges otherwise. The function is extended to all negative values, except negative integers, by the relation sx 1 1d 5 xsxd The gamma function is related to the factorial function by sn 1 1d 5 n!

n 5 1, 2, c

The solution to the wave equation is `

5

1 ` Uss, tde2isx ds 2p 30

SPECIAL FUNCTIONS

Then the complex form of the Fourier theorem is

an 5 0

1 ` Fswde2iwx dx 2p 30

for all positive integers n. An important value of the gamma function is s0.5d 5 p1>2

npx npt cos L L

Fourier transform A nonperiodic function f(x) requires two functions

Other values of the gamma function are found in CRC Standard Mathematical Tables and similar tables. Beta Function The beta function is a function of two variables and is a generalization of the binomial coefficients. It is closely related to

2-38

MATHEMATICS

the gamma function. It is defined by the integral: 1

Bsx, yd 5 3 t x21 s1 2 td y21 dt

for x, y . 0

0

The beta function can also be represented as a trigonometric integral, by substituting t  sin2 u, as p/2

Bsx, yd 5 2 3

s sin ud2x21 scos ud2y21 d u

0

The beta function is related to the gamma function by the relation Bsx, yd 5

sxdsyd sx 1 yd

This relation shows that B(x, y)  B(y, x). Bernoulli Functions The Bernoulli functions are a sequence of periodic functions of period 1 used in approximation theory. Note that for any number x, [x] represents the largest integer less than or equal to x. [3.14]  3 and [1.2]  2. The Bernoulli functions Bn(x) are defined recursively as follows: 1. B0(x)  1 2. B1(x)  x  [x]  1⁄2 3. Bn  1 is defined so that Brn11 sxd 5 Bn sxd and so that Bn  1 is periodic of period 1. B1 is a special case of the linear-sweep curve (Fig. 2.1.112). Bessel Functions of the First Kind Bessel functions of the first kind arise in the solution of Bessel’s equation of order v: x 2ys 1 xyr 1 sx 2 2 v 2dy 5 0 When this is solved using series methods, the recursive relations define the Bessel functions of the first kind of order v: `

Jv sxd 5 g k50

s21dk x v12k ¢ ≤ k!sv 1 k 1 1d 2

Chebyshev Polynomials The Chebyshev polynomials arise in the

solution of PDEs of the form s1 2 x 2dys 2 xyr 1 n2y 5 0 and in approximation theory. They are defined as follows: T0 sxd 5 1 T1 sxd 5 x

T2 sxd 5 2x 2 2 1 T3 sxd 5 4x 3 2 3x

For n  3, they are defined recursively by the relation Tn11 sxd 2 2xTn sxd 1 Tn21 sxd 5 0 Chebyshev polynomials are said to be orthogonal because they have the property 1 Tn sxdTm sxd 3 s1 2 x 2d1/2 dx 5 0 21

for n 2 m

NUMERICAL METHODS Introduction Classical numerical analysis is based on polynomial approximation of the infinite operations of integration, differentiation, and interpolation. The objective of such analyses is to replace difficult or impossible exact computations with easier approximate computations. The challenge is to make the approximate computations short enough and accurate enough to be useful. Modern numerical analysis includes Fourier methods, including the fast Fourier transform (FFT) and many problems involving the way computers perform calculations. Modern aspects of the theory are changing very rapidly. Errors Actual value  calculated value  error. There are several sources of errors in a calculation: mistakes, round-off errors, truncation errors, and accumulation errors.

Round-off errors arise from the use of a number not sufficiently accurate to represent the actual value of the number, for example, using 3.14159 to represent the irrational number p, or using 0.56 to represent 9⁄16 or 0.5625. Truncation errors arise when a finite number of steps are used to approximate an infinite number of steps, for example, the first n terms of a series are used instead of the infinite series. Accumulation errors occur when an error in one step is carried forward into another step. For example, if x  0.994 has been previously rounded to 0.99, then 1  x will be calculated as 0.01, while its true value is 0.006. An error of less than 1 percent is accumulated into an error of over 50 percent in just one step. Accumulation errors are particularly characteristic of methods involving recursion or iteration, where the same step is performed many times, with the results of one iteration used as the input for the next. Simultaneous Linear Equations The matrix equation Ax  b can be solved directly by finding A1, or it can be solved iteratively, by the method of iteration in total steps: 1. If necessary, rearrange the rows of the equation so that there are no zeros on the diagonal of A. 2. Take as initial approximations for the values of xi:

b1 x s0d 1 5 a11

b2 x s0d 2 5 a22

c

bn x s0d n 5 a nn

3. For successive approximations, take # # # 2 a x skdd/a x sk11d 5 sbi 2 ai1x skd i 1 2 in n ii Repeat step 3 until successive approximations for the values of xi reach the specified tolerance. A property of iteration by total steps is that it is self-correcting: that is, it can recover both from mistakes and from accumulation errors. Zeros of Functions An iterative procedure for solving an equation f(x)  0 is the Newton-Raphson method. The algorithm is as follows: 1. Choose a first estimate of a root x0. 2. Let xk  1  xk  f(xk)/f(xk). Repeat step 2 until the estimate xk converges to a root r. 3. If there are other roots of f(x), then let g(x)  f(x)/(x  r) and seek roots of g(x). False Position If two values x0 and x1 are known, such that f (x0) and f (x1) are opposite signs, then an iterative procedure for finding a root between x0 and x1 is the method of false position. 1. Let m  [ f (x1)  f (x0)]/(x1  x0). 2. Let x2  x1  f (x1)/m. 3. Find f(x2). 4. If f (x2) and f (x1) have the same sign, then let x1  x2. Otherwise, let x0  x2. 5. If x1 is not a good enough estimate of the root, then return to step 1. Functional Equatities To solve an equation of the form f(x)  g(x), use the methods above to find roots of the equation f(x)  g(x)  0. Maxima One method for finding the maximum of a function f(x) on an interval [a, b] is to find the roots of the derivative f (x). The maximum of f(x) occurs at a root or at an endpoint a or b. Fibonacci Search An iterative procedure for searching for maxima works if f(x) is unimodular on [a, b]. That is, f has only one maximum, and no other local maxima, between a and b. This procedure takes advantage of the so-called golden ratio, r 5 0.618034 5 s 25 2 1d/2, which arises from the Fibonacci sequence. 1. If a is a sufficiently good estimate of the maximum, then stop. Otherwise, proceed to step 2. 2. Let x1  ra  (1  r)b, and let x2  (1  r)a  rb. Note x1 x2. Find f(x1) and f (x2). a. If f (x1)  f (x2), then let a  x1 and b  x2, and go to step 1. b. If f(x1) f(x2), then let a  x1, and go to step 1. c. If f(x1)  f(x2), then let b  x2, and return to step 1.

NUMERICAL METHODS

In cases b and c, computation is saved since the new value of one of x1 and x2 will have been used in the previous step. It has been proved that the Fibonacci search is the fastest possible of the general “cutting” type of searches. Steepest Ascent If z  f(x, y) is to be maximized, then the method of steepest ascent takes advantage of the fact that the gradient, grad ( f ) always points in the direction that f is increasing the fastest. 1. Let (x0, y0) be an initial guess of the maximum of f. 2. Let e be an initial step size, usually taken to be small. 3. Let (xk  1, yk  1)  (xk, yk)  e grad f (xk, yk)/|grad f (xk, yk)|. 4. If f(xk  1, yk  1) is not greater than f (xk, yk), then replace e with e/2 (cut the step size in half ) and reperform step 3. 5. If (xk, yk) is a sufficiently accurate estimate of the maximum, then stop. Otherwise, repeat step 3. Minimization The theory of minimization exactly parallels the theory of maximization, since minimizing z  f (x) occurs at the same value of x as maximizing w   f (x). Numerical Differentiation In general, numerical differentiation should be avoided where possible, since differentiation tends to be very sensitive to small errors in the value of the function f (x). There are several approximations to f (x), involving a “step size” h usually taken to be small:

2-39

may be approximated by n

g [ f sxid 1 f sxi21d] i51

xi11 2 xi 2

If the values xi are equally spaced at distance h and if fi is written for f (xi), then the above formula reduces to h [ f0 1 2f1 1 2f2 1 # # # 1 2fn21 1 fn] 2 The error in the trapezoid rule is given by sb 2 ad3|f s std|

|E n| #

12n2

where t is some value a # t # b. Simpson’s Rule The most widely used rule for numerical integration approximates the curve with parabolas. The interval a x b must be divided into n/2 subintervals, each of length 2h, where n is an even number. Using the notation above, the integral is approximated by h [ f0 1 4 f1 1 2 f2 1 4 f3 1 # # # 1 4 fn21 1 fn] 3 The error term for Simpson’s rule is given by |E n| , nh5| f s4d std|/180, where a t b. Simpson’s rule is generally more accurate than the trapezoid rule.

fsx 1 hd 2 f sxd f rsxd 5 h f rsxd 5

f sx 1 hd 2 f sx 2 hd 2h

f rsxd 5

f sx 1 2hd 1 f sx 1 hd 2 f sx 2 hd 2 f sx 2 2hd 6h

Other formulas are possible. If a derivative is to be calculated from an equally spaced sequence of measured data, y1, y2, . . . , yn, then the above formulas may be adapted by taking yi  f(xi). Then h  xi 1  xi is the distance between measurements. Since there are usually noise or measurement errors in measured data, it is often necessary to smooth the data, expecting that errors will be averaged out. Elementary smoothing is by simple averaging, where a value yi is replaced by an average before the derivative is calculated. Examples include: yi d

yi11 1 yi 1 yi21 3

yi d

yi12 1 yi11 1 yi 1 yi21 1 yi22 5

Ordinary Differential Equations Modified Euler Method Consider a first-order differential equation dy/dx  f (x, y) and initial condition y  y0 and x  x0. Take xi equally spaced, with xi 1  xi  h. Then the method is: 1. Set n  0. 2. y nr 5 f sxn, ynd and ys 5 fx sxn, ynd 1 y nr fy sxn, ynd, where fx and fy denote partial derivatives. 3. y nr 11 5 f sxn11, yn11d. Predictor steps: 4. For n . 0, y *n11 5 yn21 1 2hy rn. * 5. y rn11 5 f sxn11, y *n11d. Corrector steps: 6. y #n 1 1 5 yn 1 [ y n11 * 1 y nr ]h/2. # 7. y rn11 5 f sxn11, y #n11d. 8. If required accuracy is not yet obtained for yn1 and y rn11, then substitute y# for y*, in all its forms, and repeat the corrector steps. Otherwise, set n  n  1 and return to step 2. Other predictor-corrector methods are described in the literature. Runge-Kutta Methods These make up a family of widely used methods for ordinary differential equations. Given dy/dx  f (x, y) and h  interval size, third-order method (error proportional to h4):

k0 5 hf sxnd

More information may be found in the literature under the topics linear filters, digital signal processing, and smoothing techniques.

k0 h k1 5 hf ¢ xn 1 , yn 1 ≤ 2 2

Numerical Integration

k2 5 hf sxn 1 h, yn 1 2k1 2 k0d

Numerical integration requires a great deal of calculation and is usually done with the aid of a computer. All the methods described here, and many others, are widely available in packaged computer software. There is often a temptation to use whatever software is available without first checking that it really is appropriate. For this reason, it is important that the user be familiar with the methods being used and that he or she ensure that the error terms are tolerably small. Trapezoid Rule If an interval a # x # b is divided into subintervals x0, x1, . . . , xn, then the definite integral b

3 f sxd dx a

yn11 5 yn 1

k0 1 4k1 1 k2 6

Higher-order Runge-Kutta methods are described in the literature. In general, higher-order methods yield smaller error terms. Partial Differential Equations

Approximate solutions to partial differential equations require even more computational effort than ordinary differential equations. A typical problem involves finding the value of a function of two or more variables on the interior of a region, given the values of the function on the boundary of that region and also given a partial differential equation that the function must satisfy.

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MATHEMATICS

'2z 1 < 2 sz1,0 2 2z0,0 1 z21,0d 'x 2 a '2z 1 < 2 sz1,1 2 z21,1 1 z21,21 2 z1,21d 'x'y 4a '2z 1 < 2 sz0,1 2 2z0,0 1 z0,21d 'y 2 a

A typical solution strategy involves representing the continuous problem by its values on a finite set of points, called nodes, within the region, and representing the differential equation by difference equations on the value of the function at the nodes. The boundary value conditions are represented by assigning values to those nodes at or near the boundary of the region. z0,2

z –2,0

z –1,1

z 0,1

z 1,1

z –1,0

z 0,0

z 1,0

z –1,–1

z 0,–1

z 1,–1

Other formulas are possible, and there are also formulas for higher order partial derivatives. Applying the appropriate difference formulas to each node in a net gives an equation for each node. If there are n nodes, then n linear equations involving n unknowns arise. Though n is likely to be rather large, the system of linear equations is sparse, that is, most of the entries in the n 3 n matrix that describes the system are zero. Special methods exist for solving sparse systems of simultaneous linear equations relatively efficiently.

z 2,0

EXAMPLE.

Solve the generalized Laplace equation =2f 5

z 0,–2

'2 f 'x

2

1

'2 f 'y 2

5 gsx, yd

where g(x, y) is a given function of x and y. At each point sx i, yjd on the interior of the region, we approximate the partial derivatives as

Fig. 2.1.113 Rectangular net.

The pattern of nodes within the region is called a net. A rectangular net is shown in Fig. 2.1.113. There is also an extensive theory of triangular nets. Some techniques use an irregular pattern of nodes, with the nodes more concentrated in more critical parts of the region. Such techniques are called finite element methods. If the distance between nodes in a net is given by a, then the various partial derivatives at a value z0,0 5 f sx0, y0d may be given by 'z 1 < sz1,0 2 z21,0d 'x 2a 'z 1 < sz0,1 2 z0,21d 'y 2a

'2z 1 < 2 szi11, j 2 2zi, j 1 zi21, jd 'x 2 a

and

'2z 1 < 2 szi, j11 2 2zi, j 1 zi, j21d. 'y 2 a

This gives a difference equation 1 sz 1 z i21, j 1 z i, j11 1 z i, j21 2 4z i, jd 5 gsx i, yjd a 2 i11, j

that must be satisfied at each point (xi, yj). Relaxation methods are a family of techniques for finding approximate solutions to such systems of equations. They involve finding the points (xi, y j) for which the difference equations are farthest from being true, and adjusting or relaxing the value of zi, j so that the equation is satisfied at that point.

2.2 COMPUTERS by Thomas J. Cockerill REFERENCES: Manuals from Computer Manufacturers. Knuth, “The Art of Computer Programming,” vols 1, 2, and 3, Addison-Wesley. Yourdon and Constantine, “Structured Design,” Prentice-Hall. DeMarco, “Structured Analysis and System Specification,” Prentice-Hall. Moshos, “Data Communications,” West Publishing. Date, “An Introduction to Database Systems,” 4th ed., Addison-Wesley. Wiener and Sincovec, “Software Engineering with Modula-2 and ADA,” Wiley. Hamming, “Numerical Methods for Scientists and Engineers,” McGraw-Hill. Bowers and Sedore, “SCEPTRE: A Computer Program for Circuit and System Analysis,” Prentice-Hall. Tannenbaum, “Operating Systems,” Prentice-Hall. Lister, “Fundamentals of Operating Systems, 3d ed., SpringerVerlag. American National Standard Programming Language FORTRAN, ANSI X3.198-1992. Jensen and Wirth, “PASCAL: User Manual and Report,” Springer. Communications, Journal, and Computer Surveys, ACM Computer Society. Computer, Spectrum, IEEE.

COMPUTER PROGRAMMING Machine Types

Computers are machines used for automatically processing information represented by mechanical, electrical, or optical means. They may be

classified as analog or digital according to the techniques used to represent and process the information. Analog computers represent information as physically measurable, continuous quantities and process the information by components that have been interconnected to form an analogous model of the problem to be solved. Digital computers, on the other hand, represent information as discrete physical states which have been encoded into symbolic formats, and process the information by sequences of operational steps which have been preplanned to solve the given problem. When compared to analog computers, digital computers have the advantages of greater versatility in solving scientific, engineering, and commercial problems that involve numerical and nonnumerical information; of an accuracy dictated by significant digits rather than that which can be measured; and of exact reproducibility of results that stay unvitiated by small, random fluctuations in the physical signals. In the past, multiple-purpose analog computers offered advantages of speed and cost in solving a sophisticated class of complex problems dealing with networks of differential equations, but these advantages have disappeared with the advances in solid-state computers. Other than the

COMPUTER DATA STRUCTURES

occasional use of analog techniques for embedding computations as part of a larger system, digital techniques now account almost exclusively for the technology used in computers. Digital information may be represented as a series of incremental, numerical steps which may be manipulated to position control devices using stepping motors. Digital information may also be encoded into symbolic formats representing digits, alphabetic characters, arithmetic numbers, words, linguistic constructs, points, and pictures which may be processed by a variety of mechanized operators. Machines organized in this manner can handle a more general class of both numerical and nonnumerical problems and so form by far the most common type of digital machines. In fact, the term computer has become synonymous with this type of machine. Digital Machines

Digital machines consist of two kinds of circuits: memory cells, which effectively act to delay signals until needed, and logical units, which perform basic Boolean operations such as AND, OR, NOT, XOR, NAND, and NOR. Memory circuits can be simply defined as units where information can be stored and retrieved on demand. Configurations assembled from the Boolean operators provide the macro operators and functions available to the machine user through encoded instructions. Both data and the instructions for processing the data can be stored in memory. Each unit of memory has an address at which the contents can be retrieved, or “read.” The read operation makes the contents at an address available to other parts of the computer without destroying the contents in memory. The contents at an address may be changed by a write operation which inserts new information after first nullifying the previous contents. Some types of memory, called read-only memory (ROM), can be read from but not written to. They can only be changed at the factory. Abstractly, the address and the contents at the address serve roles analogous to a variable and the value of the variable. For example, the equation z  x  y specifies that the value of x added to the value of y will produce the value of z. In a similar way, the machine instruction whose format might be: add, address 1, address 2, address 3 will, when executed, add the contents at address 1 to the contents at address 2 and store the result at address 3. As in the equation where the variables remain unaltered while the values of the variables may be changed, the addresses in the instruction remain unaltered while the contents at the address may change. Computers differ from other kinds of mechanical and electrical machines in that computers perform work on information rather than on forces and displacements. A common form of information is numbers. Numbers can be encoded into a mechanized form and processed by the four rules of arithmetic (  , ,  , ). But numbers are only one kind of information that can be manipulated by the computer. Given an encoded alphabet, words and languages can be formed and the computer can be used to perform such processes as information storage and retrieval, translation, and editing. Given an encoded representation of points and lines, the computer can be used to perform such functions as drawing, recognizing, editing, and displaying graphs, patterns, and pictures. Because computers have become easily accessible, engineers and scientists from every discipline have reformatted their professional activities to mechanize those aspects which can supplant human thought and decision. In this way, mechanical processes can be viewed as augmenting human physical skills and strength, and information processes can be viewed as augmenting human mental skills and intelligence. COMPUTER DATA STRUCTURES Binary Notation

Digital computers represent information by strings of digits which assume one of two values: 0 or 1. These units of information are called bits, a word contracted from the term binary digits. A string of bits may represent either numerical or nonnumerical information.

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In order to achieve efficiency in handling the information, the computer groups the bits together into units containing a fixed number of bits which can be referenced as discrete units. By encoding and formatting these units of information, the computer can act to process them. Units of 8 bits, called bytes, are common. A byte can be used to encode the basic symbolic characters which provide the computer with input-output information such as the alphabet, decimal digits, punctuation marks, and special characters. Bit groups may be organized into larger units of 4 bytes (32 bits) called words, or even larger units of 8 bytes called double words; and sometimes into smaller units of 2 bytes called half words. Besides encoding numerical information and other linguistic constructs, these units are used to encode a repertoire of machine instructions. Older machines and special-purpose machines may have other word sizes. Computers process numerical information represented as binary numbers. The binary numbering system uses a positional notation similar to the decimal system. For example, the decimal number 596.37 represents the value 5  102  9  101  6  100  3  101  7  102. The value assigned to any of the 10 possible digits in the decimal system depends on its position relative to the decimal point (a weight of 10 to zero or positive exponent is assigned to the digits appearing to the left of the decimal point, and a weight of 10 to a negative exponent is applied to digits to the right of the decimal point). In a similar manner, a binary number uses a radix of 2 and two possible digits: 0 and 1. The radix point in the positional notation separates the whole from the fractional part of the number, just as in the decimal system. The binary number 1011.011 represents a value 1  23  0  22  1  21  1  20  0  21  1  22  1  23. The operators available in the computer for setting up the solution of a problem are encoded into the instructions of the machine. The instruction repertoire always includes the usual arithmetic operators to handle numerical calculations. These instructions operate on data encoded in the binary system. However, this is not a serious operational problem, since the user specifies the numbers in the decimal system or by mnemonics, and the computer converts these formats into its own internal binary representation. On occasions when one must express a number directly in the binary system, the number of digits needed to represent a numerical value becomes a handicap. In these situations, a radix of 8 or 16 (called the octal or hexadecimal system, respectively) constitutes a more convenient system. Starting with the digit to the left or with the digit to the right of the radix point, groups of 3 or 4 binary digits can be easily converted to equivalent octal or hexadecimal digits, respectively. Appending nonsignificant 0s as needed to the rightmost and leftmost part of the number to complete the set of 3 or 4 binary digits may be necessary. Table 2.2.1 lists the conversions of binary digits to their equivalent octal and hexadecimal representations. In the hexadecimal system, the letters A through F augment the set of decimal digits to represent the digits for 10 through 15. The following examples illustrate the conversion Table 2.2.1 Binary-Hexadecimal and Binary-Octal Conversion Binary

Hexadecimal

Binary

Octal

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0 1 2 3 4 5 6 7 8 9 A B C D E F

000 001 010 011 100 101 110 111

0 1 2 3 4 5 6 7

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COMPUTERS Table 2.2.2 Schemes for Encoding Decimal Digits Decimal digit

BCD

Excess-3

4221 code

0 1 2 3 4 5 6 7 8 9

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

0011 0100 0101 0110 0111 1000 1001 1010 1011 1100

0000 0001 0010 0011 0110 1001 1100 1101 1110 1111

between binary numbers and octal or hexadecimal numbers using the table. binary number octal number

011 3

binary number hexadecimal number

011 3

110 6

101 5

. .

001 1

111 7

0110 6

1111 F

0101 5

. .

0011 3

1110 E

100 4

Formats for Numerical Data

Three different formats are used to represent numerical information internal to the computer: fixed-point, encoded decimal, and floating-point. A word or half word in fixed-point format is given as a string of 0s and 1s representing a binary number. The program infers the position of the radix point (immediately to the right of the word representing integers, and immediately to the left of the word representing fractions). Algebraic numbers have several alternate forms: 1’s complement, 2’s complement, and signed-magnitude. Most often 1’s and 2’s complement forms are adopted because they lead to a simplification in the hardware needed to perform the arithmetic operations. The sign of a 1’s complement number can be changed by replacing the 0s with 1s and the 1s with 0s. To change the sign of a 2’s complement number, reverse the digits as with a 1’s-complement number and then add a 1 to the resulting binary number. Signed-magnitude numbers use the common representation of an explicit  or  sign by encoding the sign in the leftmost bit as a 0 or 1, respectively. Many computers provide an encoded-decimal representation as a convenience for applications needing a decimal system. Table 2.2.2 gives three out of over 8000 possible schemes used to encode decimal digits in which 4 bits represent each decade value. Many other codes are possible using more bits per decade, but four bits per decimal digit are common because two decimal digits can then be encoded in one byte. The particular scheme selected depends on the properties needed by the devices in the application. The floating-point format is a mechanized version of the scientific notation ( M  10 E, where M and E represent the signed mantissa and signed exponent of the number). This format makes possible the use of a machine word to encode a large range of numbers. The signed mantissa and signed exponent occupy a portion of the word. The exponent is implied as a power of 2 or 16 rather than of 10, and the radix point is implied to the left of the mantissa. After each operation, the machine adjusts the exponent so that a nonzero digit appears in the most significant digit of the mantissa. That is, the mantissa is normalized so that its value lies in the range of 1/b # M 1 where b is the implied base of the number system (e.g.: 1/2 # M 1 for a radix of 2, and 1/16 # M 1 for a radix of 16). Since the zero in this notation has many logical representations, the format uses a standard recognizable form for zero, with a zero mantissa and a zero exponent, in order to avoid any ambiguity. When calculations need greater precision, floating-point numbers use a two-word representation. The first word contains the exponent and mantissa as in the one-word floating point. Precision is increased by appending the extra word to the mantissa. The terms single precision and double precision make the distinction between the one- and

Fig. 2.2.1 ASCII code set.

two-word representations for floating-point numbers, although extended precision would be a more accurate term for the two-word form since the added word more than doubles the number of significant digits. The equivalent decimal precision of a floating-point number depends on the number n of bits used for the unsigned mantissa and on the implied base b (binary, octal, or hexadecimal). This can be simply expressed in equivalent decimal digits p as: 0.0301 (n  log2b) p 0.0301 n. For example, a 32-bit number using 7 bits for the signed exponent of an implied base of 16, 1 bit for the sign of the mantissa, and 24 bits for the value of the mantissa gives a precision of 6.02 to 7.22 equivalent decimal digits. The fractional parts indicate that some 7-digit and some 8-digit numbers cannot be represented with a mantissa of 24 bits. On the other hand, a double-precision number formed by adding another word of 32 bits to the 24-bit mantissa gives a precision of 15.65 to 16.85 equivalent decimal digits. The range r of possible values in floating-point notation depends on the number of bits used to represent the exponent and the implied radix. For example, for a signed exponent of 7 bits and an implied base of 16, then 1664 # r # 1663. Formats of Nonnumerical Data

Logical elements, also called Boolean elements, have two possible values which simply represent 0 or 1, true or false, yes or no, OFF or ON, etc. These values may be conveniently encoded by a single bit. A large variety of codes are used to represent the alphabet, digits, punctuation marks, and other special symbols. The most popular ones are the 7-bit ASCII code and the 8-bit EBCDIC code. ASCII and EBCDIC find their genesis in punch-tape and punch-card technologies, respectively, where each character was encoded as a combination of punched holes in a column. Both have now evolved into accepted standards represented by a combination of 0s and 1s in a byte. Figure 2.2.1 shows the ASCII code. (ASCII stands for American Standard Code for Information Interchange.) The possible 128 bit patterns divide the code into 96 graphic characters (although the codes 0100000 and 1111111 do not represent any printable graphic symbol) and 32 control characters which represent nonprintable characters used in communications, in controlling peripheral machines, or in expanding the code set with other characters or fonts. The graphic codes and the control codes are organized so that subsets of usable codes with fewer bits can be formed and still maintain the pattern. Data Structure Types

The above types of numerical and nonnumerical data formats are recognized and manipulated by the hardware operations of the computer. Other more complex data structures may be programmed into the

COMPUTER ORGANIZATION

computer by building upon these primitive data types. The programmable data structures might include arrays, defined as ordered lists of elements of identical type; sets, defined as unordered lists of elements of identical type; records, defined as ordered lists of elements that need not be of the same type; files, defined as sequential collections of identical records; and databases, defined as organized collections of different records or file types. COMPUTER ORGANIZATION

the premise that data and instructions that will shortly be needed are located near those currently being used. If the information is not found in the cache, then it is transferred from the main memory. The effective average access times offered by the combined configuration of RAM and cache results in a more powerful (faster) computer. Central Processing Unit

The CPU makes available a repertoire of instructions which the user uses to set up the problem solutions. Although the specific format for instructions varies among machines, the following illustrates the pattern:

Principal Components

The principal components of a computer system consist of a central processing unit (referred to as the CPU or platform), its working memory, user interface, file storage, and a collection of add-ons and peripheral devices. A computer system can be viewed as a library of collected data and packages of assembled sequences of instructions that can be executed in the prescribed order by the CPU to solve specific problems or perform utility functions for the users. These sequences are variously called programs, subprograms, routines, subroutines, procedures, functions, etc. Collectively they are called software and are directly accessible to the CPU through the working memory. The file devices act analogously to a bookshelf—they store information until it is needed. Only after a program and its data have been transferred from the file devices or from peripheral devices to the working memory can the individual instructions and data be addressed and executed to perform their intended functions. The CPU functions to monitor the flow of data and instructions into and out of memory during program execution, control the order of instruction execution, decode the operation, locate the operand(s) needed, and perform the operation specified. Two characteristics of the memory and storage components dictate the roles they play in the computer system. They are access time, defined as the elapsed time between the instant a read or write operation has been initiated and the instant the operation is completed, and size, defined by the number of bytes in a module. The faster the access time, the more costly per bit of memory or storage, and the smaller the module. The principal types of memory and storage components from the fastest to the slowest are registers which operate as an integral part of the CPU, cache and main memory which form the working memory, and mass and archival storage which serve for storing files. The interrelationships among the components in a computer system and their primary performance parameters will be given in context in the following discussion. However, hundreds of manufacturers of computers and computer products have a stake in advancing the technology and adding new functionality to maintain their competitive edge. In such an environment, no performance figures stay current. With this caveat, performance figures given should not be taken as absolutes but only as an indication of how each component contributes to the performance of the total system. Throughout the discussion (and in the computer world generally), prefixes indicating large numbers are given by the symbols k for kilo (103), M for mega (106), G for giga (109), and T for tera (1012). For memory units, however, these symbols have a slightly altered meaning. Memories are organized in binary units whereby powers of two form the basis for all addressing schemes. According, k refers to a memory size of 1024 (210) units. Similarly M refers to 10242 (1,048,576), G refers to 10243, and T refers to 10244. For example, 1-Mbyte memory indicates a size of 1,048,576 bytes. Memory

The main memory, also known as random access memory (RAM), is organized into fixed size bit cells (words, bytes, half words, or double words) which can be located by address and whose contents contain the instructions and data currently being executed. The CPU acts to address the individual memory cells during program execution and transfers their contents to and from its internal registers. Optionally, the working memory may contain an auxiliary memory, called cache, which is faster than the main memory. Cache operates on

2-43

name: operator, operand(s) The name designates an address whose contents contain the operator and one or more operands. The operator encodes an operation permitted by the hardware of the CPU. The operand(s) refer to the entities used in the operation which may be either data or another instruction specified by address. Some instructions have implied operand(s) and use the bits which would have been used for operand(s) to modify the operator. To begin execution of a program, the CPU first loads the instructions serially by address into the memory either from a peripheral device, or more frequently, from storage. The internal structure of the CPU contains a number of memory registers whose number, while relatively few, depend on the machine’s organization. The CPU acts to transfer the instructions one at a time from memory into a designated register where the individual bits can be interpreted and executed by the hardware. The actions of the following steps in the CPU, known as the fetch-execute cycle, control the order of instruction execution. Step 1: Manually or automatically under program control load the address of the starting instruction into a register called the program register (PR). Step 2: Fetch and copy the contents at the address in PR into a register called the program content register (PCR). Step 3: Prepare to fetch the next instruction by augmenting PR to the next address in normal sequence. Step 4: Interpret the instruction in PCR, retrieve the operands, execute the encoded operation, and then return to step 2. Note that the executed instruction may change the address in PR to start a different instruction sequence. The speed of machines can be compared by calculating the average execution time of an instruction. Table 2.2.3 illustrates a typical instruction mix used in calculating the average. The instruction mix gives the relative frequency each instruction appears in a compiled list of typical programs and so depends on the types of problems one expects the machine to solve (e.g., scientific, commercial, or combination). The equation t 5 g witi i

expresses the average instruction execution time t as a function of the execution time ti for instruction i having a relative frequency wi in the instruction mix. The reciprocal of t measures the processor’s performance as the average number of instructions per second (ips) it can execute. Table 2.2.3

Instruction Mix

i

Instruction type

Weight wi

1 2 3 4 5 6 7 8

Add: Floating point Fixed point Multiple: Floating point Load/store register Shift: One character Branch: Conditional Unconditional Move 3 words in memory Total

0.07 0.16 0.06 0.12 0.11 0.21 0.17 0.10 1.00

For machines designed to support scientific and engineering calculations, the floating-point arithmetic operations dominate the time needed to execute an average instruction mix. The speed for such machines is

2-44

COMPUTERS

given by the average number of floating-point operations which can be executed per second (flops). This measure, however, can be misleading in comparing different machine models. For example, when the machine has been configured with a cluster of processors cooperating through a shared memory, the rate of the configuration (measured in flops) represents the simple sum of the individual processors’ flops rates. This does not reflect the amount of parallelism that can be realized within a given problem. To compare the performance of different machine models, users often assemble and execute a suite of programs which characterize their particular problem load. This idea has been refined so that in 1992 two suites of benchmark programs representing typical scientific, mathematical, and engineering applications were standardized: Specint92 for integer operations, and Specfp92 for floating-point operations. Since these benchmark programs were established, many more standardized benchmarks have been created to model critical workloads and applications, and the set is constantly being revised and expanded. Performance ratings for computers are often reported on the basis of results from running a selected subset of the available benchmark programs. Computer performance depends on a number of interrelated factors used in their design and fabrication, among them compactness, bus size, clock frequency, instruction set size, and number of coprocessors. The speed that energy can be transmitted through a wire, bounded theoretically at 3  1010 cm/s, limits the ultimate speed at which the electronic circuits can operate. The further apart the electronic elements are from each other, the slower the operations. Advances in integrated circuits have produced compact microprocessors operating in the nanosecond range. The microprocessor’s bus size (the width of its data path, or the number of bits that can be sent simultaneously in parallel) affect its performance in two ways: by the number of memory cells that can be directly addressed, and by the number of bits each memory reference can fetch and process at a time. For example, a 16-bit microprocessor can reference 216 16-bit memory cells and process 16 bits at a time. In order to handle the individual bits, the number of transistors that must be packed into the microprocessor goes up geometrically with the width of the data path. The earliest microprocessors were 8-bit devices, meaning that every memory reference retrieved 8 bits. To retrieve more bits, say 16, 32, or 64 bits, the 8-bit microprocessor had to make multiple references. Microprocessors have become more powerful as the packing technology has improved. While normally the circuits operate asynchronously, a computer clock times the sequencing of the instructions. Clock speed is given in hertz (Hz, one cycle per second). Each instruction takes an integral number of cycles to complete, with one cycle being the minimum. If an instruction completes its operations in the middle of a cycle, the start of the next instruction must wait for the beginning of the next cycle. Two schemes are used to implement the computer instruction set in the microprocessors. The more traditional complex instruction set computer (CISC) microprocessors implement by hard-wiring some 300 instruction types. Strange to say, the faster alternate-approach reduced instruction set computer (RISC) implements only about 10 to 30 percent of the instruction types by hard wiring, and implements the remaining more-complex instructions by programming them at the factory into read-only memory. Since the fewer hard-wired instructions are more frequently used and can operate faster in the simpler, more-compact RISC environment, the average instruction time is reduced. To achieve even greater effectiveness and speed calls for more complex coordination in the execution of instructions and data. In one scheme, microprocessors in a cluster share a common memory to form the machine organization (a multiprocessor or parallel processor). The total work which may come from a single program or independent programs is parceled out to the individual machines which operate independently but are coordinated to work in parallel with the other machines in the cluster. Faster speeds can be achieved when the individual processors can work on different parts of the problem or can be assigned to those parts of the problem solution for which they have been especially designed (e.g., input-output operations or computational

operations). Two other schemes, pipelining and array processing, divide an instruction into the separate tasks that must be performed to complete its execution. A pipelining machine executes the tasks concurrently on consecutive pieces of data. An array processor executes the tasks of the different instructions in a sequence simultaneously and coordinates their completion (which might mean abandoning a partially completed instruction if it had been initiated prematurely). These schemes are usually associated with the larger and faster machines. User Interface

The user interface is provided to enable the computer user to initiate or terminate tasks, to interrogate the computer to determine the status of tasks during execution, to give and receive instructions, and to otherwise monitor the operation of the system. The user interface is continuously evolving, but commonly consists of a relatively slow speed keyboard input, a display, and a pointing device. Important display characteristics for usability are the size of the screen and the resolution. The display has its own memory, which refreshes and controls the display. Examples of display devices that have seen common usage are cathode-ray tube (CRT) displays, flatpanel displays, and thin-film displays. For convenience and manual speed, a pointing device such as a mouse, trackball, or touchpad can be used to control the movement of the cursor on the display. The pointing device can also be used to locate and select options available on the display. File Devices

File devices serve to store libraries of directly accessible programs and data in electronically or optically readable formats. The file devices record the information in large blocks rather than by individual addresses. To be used, the blocks must first be transferred into the working memory. Depending on how selected blocks are located, file devices are categorized as sequential or direct-access. On sequential devices the computer locates the information by searching the file from the beginning. Direct-access devices, on the other hand, position the read-write mechanism directly at the location of the needed information. Searching on these devices works concurrently with the CPU and the other devices making up the computer configuration. Magnetic or optical disks that offer a wide choice of options form the commercial direct-access devices. Some file devices are read only and are used to deliver programs or data to be installed or only referenced by the user. Examples of read-only devices are CD-ROM and DVD-ROM. Other file devices can be both read from and written to. Example of read-write devices are magnetic disk drives, CD RJW drives, and DVD-RW drives. The recording surface consists of a platter (or platters) of recording material mounted on a common spindle rotated at high speed. The read-write heads may be permanently positioned along the radius of the platter or may be mounted on a common arm that can be moved radially to locate any specified track of information. Information is recorded on the tracks circumferentially using fixed-size blocks called pages or sectors. Pages divide the storage and memory space alike into blocks of 4096 bytes so that program transfers can be made without creating unusable space. Sectors nominally describe the physical division of the storage space into equal segments for easier positioning of the read-write heads. The access time for retrieving information from a disk depends on three separately quoted factors, called seek time, latency time, and transfer time. Seek time gives the time needed to position the read-write heads from their current track position to the track containing the information. Since the faster fixed-head disks require no radial motion, only latency and transfer time need to be factored into the total access time for these devices. Latency time is the time needed to locate the start of the information along the circumferential track. This time depends on the speed of revolution of the disk, and, on average, corresponds to the time required to revolve the platter half a turn. The average latency time can be reduced by repeating the information several times around the track. Transfer time, usually quoted as a rate, gives the rate at which information can be transferred to memory after it has been located. There is a large variation in transfer rates depending on the disk system selected.

DISTRIBUTED COMPUTING

Computer architects sometimes refer to file storage as mass storage or archival storage, depending on whether or not the libraries can be kept off-line from the system and mounted when needed. Disk drives with mountable platter(s) and tape drives constitute the archival storage. Sealed disks that often have fixed heads for faster access are the medium of choice for mass storage. Peripheral Devices and Add-ons

Peripheral devices function as self-contained external units that work on line to the computer to provide or receive information or to control the flow of information. Add-ons are a special class of units whose circuits can be integrated into the circuitry of the computer hardware to augment the basic functionality of the processors. Section 15 covers the electronic technology associated with these devices. An input device may be defined as any device that provides a machine-readable source of information. For engineering work, the most common forms of input are touch-tone dials, mark sensing, bar codes, and keyboards (usually in conjunction with a printing mechanism or video scope). Many bench instruments have been reconfigured to include digital devices to provide direct input to computers. Because of the datahandling capabilities of the computer, these instruments can be simpler, smaller, and less expensive than the hand instruments they replace. New devices have also been introduced: devices for visual measurement of distance, area, speed, and coordinate position of an object; or for inspecting color or shades of gray for computer-guided vision. Other methods of input that are finding greater acceptance include handwriting recognition, printed character recognition, voice digitizers, and picture digitizers. Traditionally, output devices play the role of producing displays for the interpretation of results. A large variety of printers, graphical plotters, video displays, and audio sets have been developed for this purpose. A variety of actuators have been developed for driving control mechanisms. For complex numerical control, programmable controllers (called PLCs) can simultaneously control and update data from multiple tasks. These electronically driven mechanisms and controllers, working with input devices, make possible systems for automatic testing of products, real-time control, and robotics with learning and adaptive capabilities.

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workstation, one can expect a machine designed as a workstation to offer higher performance than a PC and to support the more specialized peripherals and sophisticated professional software. Nevertheless, the boundary between PCs and workstations changes as the technology advances. Notebook PCs and the smaller sized palmtop PCs are portable, battery-operated machines. These machines find excellent use as portable PCs in some applications and as data acquisition systems. However their undersized keyboards and small displays may limit their usefulness for sustained operations. Computers larger than a PC or a workstation, called mainframes (and sometimes minis or maxis, depending on size), serve to support multiusers and multiapplications. A remotely accessible computing center may house several mainframes which either operate alone or cooperate with each other. Their high speed, large memories, and high reliability allow them to handle complex programs and they have been especially well suited to applications that do not allow for any significant downtime such as banking operations, business transactions, and reservation systems. At the upper extreme end of the computer spectrum is the supercomputer, the class of the fastest machines that can address large, complex scientific/engineering problems which cannot reasonably be transferred to other machines. Obviously this class of computer must have cache and main memory sizes and speeds commensurate with the speed of the platform. While mass memory sizes must also be large, computers which support large databases may often have larger memories than do supercomputers. Large, complex technical problems must be run with high-precision arithmetic. Because of this, performance is measured in double-precision flops. To realize the increasing demand for higher performance, the designers of supercomputers work at the edge of available technology, especially in the use of multiple processors. With multiple processors, however, performance depends as much on the time spent in communication between processors as on the computational speed of the individual processors. In the final analysis, to muster the supercomputer’s inherent speed, the development of the software becomes the problem. Some users report that software must often be hand-tailored to the specific problem. The power of the machines, however, can replace years of work in analysis and experimentation.

Computer Sizes

DISTRIBUTED COMPUTING

Computer size refers not only to the physical size but also to the number of electronics elements in the system, and so reflects the performance of the system. Between the two ends of the spectrum from the largest and fastest to the smallest and slowest are machines that vary in speed and complexity. Although no nomenclature has been universally adopted that indicates computer size, the following descriptions illustrate a few generally understood terms used for some common configurations. The choice of which computer is appropriate often requires serious evaluation. In the cases where serious evaluation is required, it is necessary to evaluate common performance benchmarks for the machine. In addition, it may be necessary to evaluate the machine by using a mix of applications designed to simulate current usage of existing machines or the expected usage of the proposed machine. Personal computers (PCs) have been made possible by the advances in solid-state technology. The name applies to computers that can fit the total complement of hardware on a desktop and operate as stand-alone systems so as to provide immediate dedicated services to an individual user. This popular computer size has been generally credited for spreading computer literacy in today’s society. Because of its commercial success, many peripheral devices, add-ons, and software products have been (and are continually being) developed. Laptop PCs are personal computers that have the low weight and size of a briefcase and can easily be transported when peripherals are not immediately needed. The term workstation describes computer systems which have been designed to support complex engineering, scientific, or business applications in a professional environment. Although a top-of-the-line PC or a PC connected as a peripheral to another computer can function like a

Organization of Data Facilities

A distributed computer system can be defined as a collection of computer resources which are remotely located from each other and are interconnected to cooperate in providing their respective services. The resources include both the equipment and the software. Resources distributed to reside near the vicinity where the data is collected or used have an obvious advantage over centralization. But to provide information in a timely and reliable manner, these islands of automation must be integrated. The size and complexity of an enterprise served by a distributed information system can vary from a single-purpose office to a multipleplant conglomerate. An enterprise is defined as a system which has been created to accomplish a mission in its environment and whose goals involve risk. Internally it consists of organized functions and facilities which have been prepared to provide its services and accomplish its mission. When stimulated by an external entity, the enterprise acts to produce its planned response. An enterprise must handle both the flow of material (goods) and the flow of information. The information system tracks the material in the material system, but itself handles only the enterprise’s information. The technology for distributing and integrating the total information system comes under the industrial strategy known as computer-integrated business (CIB) or computer-integrated manufacturing (CIM). The following reasons have been cited for developing CIB and CIM: Most data generated locally has only local significance. Data integrity resides where it is generated.

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COMPUTERS

The quality and consistency of operational decisions demands not only that all parts of the system work with the same data but that they can receive it in a reliable and timely manner. If a local processor fails, it may disrupt local operations, but the remaining system should continue to function independently. Small cohesive processors can be best managed and maintained locally. Through standards, selection of local processes can be made from the best products in a competitive market that can be integrated into the total system. Obsolete processors can be replaced by processors implemented by more advance technology that conform to standards without the cost of tailoring the products to the existing system. Figure 2.2.2 depicts the total information system of an enterprise. The database consists of the organized collection of data the processors use in their operations. Because of differences in their communication requirements, the automated procedures are shown separated into those used in the office and those used on the production floor. In a business environment, the front office operations and back office operations make this separation. While all processes have a critical deadline, the production floor handles real-time operations, defined as processes which must complete their response within a critical deadline or else the results of the operations become moot. This places different constraints on the local-area networks (LANs) that serve the communication needs within the office or within the production floor. To communicate with entities outside the enterprise, the enterprise uses a wide-area network (WAN), normally made up from available public network facilities. For efficient and effective operation, the processes must be interconnected by the communications to share the data in the database and so integrate the services provided.

noise inherent in the components. The formula: C 5 W log 2 s1 1 S/Nd gives the capacity C in bits/s in terms of the signal to noise ratio S/N and the bandwidth W. Since the signal to noise ratio is normally given in decibels divisible by 3 (e.g., 12, 18, 21, 24) the following formula provides a workable approximation to the formula above: C 5 WsS/Nddb/3 where (S/N)db is the signal-to-noise ratio expressed in decibels. Other forms of noise, signal distortions, and the methods of signal modulation reduce this theoretical capacity appreciably. Nominal transmission speeds for electronic channels vary from 1000 bits to almost 20 Mbits per second. Fiber optics, however, form an almost noise-free medium. The transmission speed in fiber optics depends on the amount a signal spreads due to the multiple reflected paths it takes from its source to its destination. Advances in fiber technology have reduced this spread to give unbelievable rates. Effectively, the speeds available in today’s optical channels make possible the transmission over a common channel, using digital techniques, of all forms of information: text, voice, and pictures. Besides agreeing on speed, the transmitter and receiver must agree on the mode of transmission and on the timing of the signals. For stations located remotely from each other, transmission occurs by organizing the bits into groups and transferring them, one bit after another, in a serial mode. One scheme, called asynchronous or start-stop transmission, uses separate start and stop signals to frame a small group of bits representing a character. Separate but identical clocks at the transmitter and receiver time the signals. For transmission of larger blocks at faster rates, the stations use synchronous transmission which embeds the clock information within the transmitted bits. Communication Layer Model

Figure 2.2.3 depicts two remotely located stations that must cooperate through communication in accomplishing their respective tasks. The communications substructure provides the communication services needed by the application. The application tasks themselves, however, are outside the scope of the communication substructure. The distinction here is similar to that in a telephone system which is not concerned with the application other than to provide the needed communication service. The figure shows the communication facilities packaged into a hierarchical modular layer architecture in which each node contains identical kinds of functions at the same layer level. The layer functions represent abstractions of real facilities, but need not represent specific hardware or software. The entities at a layer provide communication Fig. 2.2.2 Composite view of an enterprise’s information system. Communication Channels

A communication channel provides the connecting path for transmitting signals between a computing system and a remotely located application. Physically the channel may be formed by a wire line using copper, coaxial cable, or optical-fiber cable; or may be formed by a wireless line using radio, microwave, or communication satellites; or may be a combination of these lines. Capacity, defined as the maximum rate at which information can be transmitted, characterizes a channel independent of the morphic line. Theoretically, an ideal noiseless channel that does not distort the signals has a channel capacity C given by: C  2W where C is in pulses per second and W is the channel bandwidth. For digital transmission, the Hartley-Shannon theorem sets the capacity of a channel limited by the presence of gaussian noise such as the thermal

Fig. 2.2.3 Communication layer architecture.

RELATIONAL DATABASE TECHNOLOGY

services to the layer above or can request the services available from the layer below. The services provided or requested are available only at service points which are identified by addresses at the boundaries that interface the adjacent layers. The top and bottom levels of the layered structure are unique. The topmost layer interfaces and provides the communication services to the noncommunication functions performed at a node dealing with the application task (the user’s program). This layer also requests communication services from the layer below. The bottom layer does not have a lower layer through which it can request communication services. This layer acts to create and recognize the physical signals transmitted between the bottom entities of the communicating partners (it arranges the actual transmission). The medium that provides the path for the transfer of signals (a wire, usually) connects the service access points at the bottom layers, but itself lies outside the layer structure. Virtual communication occurs between peer entities, those at the same level. Peer-to-peer communication must conform to layer protocol, defined as the rules and conventions used to exchange information. Actual physical communication proceeds from the upper layers to the bottom, through the communication medium (wire), and then up through the layer structure of the cooperating node. Since the entities at each layer both transmit and receive data, the protocol between peer layers controls both input and output data, depending on the direction of transmission. The transmitting entities accomplish this by appending control information to each data unit that they pass to the layer below. This control information is later interpreted and removed by the peer entities receiving the data unit. Communication Standards

Table 2.2.4 lists a few of the hundreds of forums seeking to develop and adopt voluntary standards or to coordinate standards activities. Often users establish standards by agreement that fixes some existing practice. The ISO, however, has described a seven-layer model, called the Reference Model for Open Systems Interconnection (OSI), for coordinating and expediting the development of new implementation standards. The term open systems refers to systems that allow devices to be interconnected and to communicate with each other by conforming to common implementation standards. The ISO model is not of itself an implementation standard nor does it provide a basis for appraising existing implementations, but it partitions the communication facilities into layers of related function which can be independently standardized by different teams of experts. Its importance lies in the fact that both vendors and users have agreed to provide and accept implementation standards that conform to this model. Table 2.2.4 Standards CCITT ISO ANSI EIA IEEE MAP/TOP NIST

Some Groups Involved with Communication Comité Consultatif de Télégraphique et Téléphonique International Organization for Standardization American National Standards Institute Electronic Industries Association Institute of Electrical and Electronics Engineers Manufacturing Automation Protocols and Technical and Office Protocols Users Group National Institute of Standards and Technology

The following lists the names the ISO has given the layers in its ISO model together with a brief description of their roles. Application layer provides no services to the other layers but serves as the interface for the specialized communication that may be required by the actual application, such as file transfer, message handling, virtual terminal, or job transfer. Presentation layer relieves the node from having to conform to a particular syntactical representation of the data by converting the data formats to those needed by the layer above. Session layer coordinates the dialogue between nodes including arranging several sessions to use the same transport layer at one time.

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Transport layer establishes and releases the connections between peers to provide for data transfer services such as throughput, transit delays, connection setup delays, error rate control, and assessment of resource availability. Network layer provides for the establishment, maintenance, and release of the route whereby a node directs information toward its destination. Data link layer is concerned with the transfer of information that has been organized into larger blocks by creating and recognizing the block boundaries. Physical layer generates and detects the physical signals representing the bits, and safeguards the integrity of the signals against faulty transmission or lack of synchronization. The IEEE has formulated several implementation standards for office or production floor LANs that conform to the lower two layers of the ISO model. The functions assigned to the ISO data link layer have been distributed over two sublayers, a logical link control (LLC) upper sublayer that generates and interprets the link control commands, and a medium access control (MAC) lower sublayer that frames the data units and acquires the right to access the medium. From this structure, the IEEE has formulated standards for the MAC sublayer and ISO physical layer combination, and a common standard for the LLC sublayer. CSMA/CD standardized the access method developed by the Xerox Corporation under its trademark Ethernet. The nodes in the network are attached to a common bus. All nodes hear every message transmitted, but accept only those messages addressed to themselves. When a node has a message to transmit, it listens for the line to be free of other traffic before it initiates transmission. However, more than one node may detect the free line and may start to transmit. In this situation the signals will collide and produce a detectable change in the energy level present in the line. Even after a station detects a collision it must continue to transmit to make sure that all stations hear the collision (all data frames must be of sufficient length to be present simultaneously on the line as they pass each station). On hearing a collision, all stations that are transmitting wait a random length of time and then attempt to retransmit. The MAP/TOP (Manufacturing Automation Protocols and Technical and Office Protocols) Users Group started under the auspices of General Motors and Boeing Information Systems and now has a membership of many thousands of national and international corporations. The corporations in this group have made a commitment to open systems that will allow them to select the best products through standards, agreed to by the group, that will meet their respective requirements. These standards have also been adopted by NIST for governmentwide use under the title Government Open Systems Interconnections Profile (GOSIP). The common carriers who offer WAN communication services through their public networks have also developed packet-switching networks for public use. Packet switching transmits data in a purely digital format, which, when embellished, can replace the common circuitswitching technology used in analog communications such as voice. A packet is a fixed-sized block of digital data with embedded control information. The network serves to deliver the packets to their destination in an efficient and reliable manner. Wireless network capabilities for computers based on the IEEE 802.11 set of standards evolved rapidly. The 802.11 standards enabled the implementation of wireless LAN having a performance comparable to that of wired LAN. The wireless LAN network is a set of wireless access points attached to a LAN that allow computers access the LAN. The coverage of the wireless access points can overlap allowing mobility within a region of coverage and allowing transfer from one access point to another if the computer is moved from one coverage region to another.

RELATIONAL DATABASE TECHNOLOGY Design Concepts

As computer hardware has evolved from small working memories and tape storage to large working memories and large disk storage, so has database technology moved from accessing and processing of a single,

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COMPUTERS

sequential file to that of multiple, random-access files. A relational database can be defined as an organized collection of interconnected tables or records. The records appear like the flat files of older technology. In each record the information is in columns (fields) which identify attributes, and rows (tuples) which list particular instances of the attributes. One column (or more), known as the primary key, identifies each row. Obviously, the primary key must be unique for each row. If the data is to be handled in an efficient and orderly way, the records cannot be organized in a helter-skelter fashion such as simply transporting existing flat files into relational tables. To avoid problems in maintaining and using the database, redundancy should be eliminated by storing each fact at only one place so that, when making additions or deletions, one need not worry about duplicates throughout the database. This goal can be realized by organizing the records into what is known as the third normal form. A record is in the third normal form if and only if all nonkey attributes are mutually independent and fully dependent on the primary key. The advantages of relational databases, assuming proper normalization, are: Each fact can be stored exactly once. The integrity of the data resides locally, where it is generated and can best be managed. The tables can be physically distributed yet interconnected. Each user can be given his/her own private view of the database without altering its physical structure. New applications involving only a part of the total database can be developed independently. The system can be automated to find the best path through the database for the specified data. Each table can be used in many applications by employing simple operators without having to transfer and manipulate data superfluous to the application. A large, comprehensive system can evolve from phased design of local systems. New tables can be added without corrupting everyone’s view of the data. The data in each table can be protected differently for each user (read-only, write-only). The tables can be made inaccessible to all users who do not have the right to know. Relational Database Operators

A database system contains the structured collection of data, an on-line catalog and dictionary of data items, and facilities to access and use the data. The system allows users to: Add new tables Remove old tables Insert new data into existing tables Delete data from existing tables Retrieve selected data Manipulate data extracted from several tables Create specialized reports

Table 2.2.5

As might be expected, these systems include a large collection of operators and built-in functions in addition to those normally used in mathematics. Because of the similarity between database tables and mathematical sets, special set-like operators have been developed to manipulate tables. Table 2.2.5 lists seven typical table operators. The list of functions would normally also include such things as count, sum, average, find the maximum in a column, and find the minimum in a column. A rich collection of report generators offers powerful and flexible capabilities for producing tabular listings, text, graphics (bar charts, pie charts, point plots, and continuous plots), pictorial displays, and voice output. SOFTWARE ENGINEERING Programming Goals

Software engineering encompasses the methodologies for analyzing program requirements and for structuring programs to meet the requirements over their life cycle. The objectives are to produce programs that are: Well documented Easily read Proved correct Bug- (error-) free Modifiable and maintainable Implementable in modules Control-Flow Diagrams

A control-flow diagram, popularly known as a flowchart, depicts all possible sequences of a program during execution by representing the control logic as a directed graph with labeled nodes. The theory associated with flowcharts has been refined so that programs can be structured to meet the above objectives. Without loss of generality, the nodes in a flowchart can be limited to the three types shown in Fig. 2.2.4. A function may be either a transformer which converts input data values into output data values or a transducer which converts that data’s morphological form. A label placed in the rectangle specifies the function’s action. A predicate node acts to bifurcate the path through the node. A question labels the diamond representing a predicate node. The answer to the question yields a binary value: 0 or 1, yes or no, ON or OFF. One of the output lines is selected accordingly. A connector serves to rejoin separated paths. Normally the circle representing a connector does not contain a label, but when the flowchart is used to document a computer program it may be convenient to label the connector. Structured programming theory models all programs by their flowcharts by placing minor restrictions on their lines and nodes. Specifically, a flowchart is called a proper program if it has precisely one input and one output line, and for every node there exists a path from the input line through the node to the output line. The restriction prohibiting multiple input or output lines can easily be circumvented by funneling the lines through collector nodes. The other restriction simply discards unwanted program structures, since a program with a path that does not reach the output may not terminate.

Relational Database Operators

Operator

Input

Output

Select Project Union Intersection Difference Join

A table and a condition A table and an attribute Two tables Two tables Two tables Two tables and a condition

Divide

A table, two attributes, and list of values

A table of all tuples that satisfy the given condition A table of all values in the specified attribute A table of all unique tuples appearing in one table or the other A table of all tuples the given tables have in common A table of all tuples appearing in the first and not in the second table A table concatenating the attributes of the tuples that satisfy the given condition A table of values appearing in one specified attribute of the given table when the table has tuples that satisfies every value in the list in the other given attribute

SOFTWARE ENGINEERING

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Fig. 2.2.4 Basic flowchart nodes.

Not all proper programs exhibit the desirable properties of meeting the objectives listed above. Figure 2.2.5 lists a group of proper programs whose graphs have been identified as being well-structured and useful as basic building blocks for creating other well-structured programs. The name assigned to each of these graph suggests the process each represents. CASE is just a convenient way of showing multiple IFTHENELSEs more compactly.

Fig. 2.2.6 Illustration of a control-flow diagram.

Figure 2.2.7 shows the four basic elements used to construct a data-flow diagram. The roles each element plays in the system are:

Fig. 2.2.5 Basic flowchart building blocks.

The structured programming theorem states: any proper program can be reconfigured to an equivalent program producing the same transformation of the data by a flowchart containing at most the graphs labeled BLOCK, IFTHENELSE, and REPEATUNTIL. Every proper program has one input line and one output line like a function block. The synthesis of more complex well-structured programs is achieved by substituting any of the three building blocks mentioned in the theorem for a function node. In fact, any of the basic building blocks would do just as well. A program so structured will appear as a block of function nodes with a top-down control flow. Because of the top-down structure, the arrow points are not normally shown. Figure 2.2.6 illustrates the expansion of a program to find the roots of ax2  bx  c  0. The flowchart is shown in three levels of detail.

Rectangular boxes lie outside the system and represent the input data sources or output data sinks that communicate with the system. The sources and sinks are also called terminators. Circles (bubbles) represent processes or actions performed by the system in accomplishing its function. Twin parallel lines represent a data file used to collect and store data from among the processes or from a process over time which can be later recalled. Arcs or vectors connect the other elements and represent data flows. A label placed with each element makes clear its role in the system. The circles contain verbs and the other elements contain nouns. The arcs tie the system together. An arc between a terminator and a process represents input to or output from the system. An arc between two processes represents output from one process which is input to the other. An arc between a process and a file represents data gathered by the process and stored in the file, or retrieval of data from the file. Analysis starts with a contextual view of the system studied in its environment. The contextual view gives the name of the system, the collection of terminators, and the data flows that provide the system inputs and outputs; all accompanied by a statement of the system objective. Details on the terminators and data they provide may also be described by text, but often the picture suffices. It is understood that the form of the input and output may not be dictated by the designer since they often involve organizations outside the system. Typical inputs in

Data-Flow Diagrams Data-flow diagrams structure the actions of a program into a network

by tracking the data as it passes through the program. They depict the interworkings of a system by the processes performing the work and the communication between the processes. Data-flow diagrams have proved valuable in analyzing existing or new systems to determine the system requirements and in designing systems to meet those requirements.

Fig. 2.2.7 Data-flow diagram elements.

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COMPUTERS SOFTWARE SYSTEMS Software Techniques

Fig. 2.2.8 Illustration of a data-flow diagram.

industrial systems include customer orders, payment checks, purchase orders, requests for quotations, etc. Figure 2.2.8a illustrates a context diagram for a repair shop. Figure 2.2.8b gives many more operational details showing how the parts of the system interact to accomplish the system’s objectives. The designer can restructure the internal processors and the formats of the data flows. The bubbles in a diagram can be broken down into further details to be shown in another data-flow diagram. This can be repeated level after level until the processes become manageable and understandable. To complete the system description, each bubble in the dataflow charts is accompanied by a control-flow diagram or its equivalent to describe the algorithm used to accomplish the actions and a data dictionary describing the items in the data flows and in the databases. The techniques of data-flow diagrams lend themselves beautifully to the analysis of existing systems. In a complex system it would be unusual for an individual to know all the details, but all system participants know their respective roles: what they receive, whence they receive it, what they do, what they send, and where they send it. By carefully structuring interviews, the complete system can be synthesized to any desired level of detail. Moreover, each system component can be verified because what is sent from one process must be received by another and what is received by a process must be used by the process. To automate the total system or parts of the system, control bubbles containing transition diagrams can be implemented to control the timing of the processes.

Two basic operations form the heart of nonnumerical techniques such as those found in handling large database tables. One basic operation, called sorting, collates the information in a table by reordering the items by their key into a specified order. The other basic operation, called searching, seeks to find items in a table whose keys have the same or related value as a given argument. The search operation may or may not be successful, but in either case further operations follow the search (e.g., retrieve, insert, replace). One must recognize that computers cannot do mathematics. They can perform a few basic operations such as the four rules of arithmetic, but even in this case the operations are approximations. In fact, computers represent long integers, long rationals, and all the irrational numbers like p and e only as approximations. While computer arithmetic and the computer representation of numbers exceed the precision one commonly uses, the size of problems solved in a computer and the number of operations that are performed can produce misleading results with large computational errors. Since the computer can handle only the four rules of arithmetic, complex functions must be approximated by polynomials or rational fractions. A rational fraction is a polynomial divided by another polynomial. From these curve-fitting techniques, a variety of weightedaverage formulas can be developed to approximate the definite integral of a function. These formulas are used in the procedures for solving differential and integral equations. While differentiation can also be expressed by these techniques, it is seldom used, since the errors become unacceptable. Taking advantage of the machine’s speed and accuracy, one can solve nonlinear equations by trial and error. For example, one can use the Newton-Raphson method to find successive approximations to the roots of an equation. The computer is programmed to perform the calculations needed in each iteration and to terminate the procedure when it has converged on a root. More sophisticated routines can be found in the libraries for finding real, multiple, and complex roots of an equation. Matrix techniques have been commercially programmed into libraries of prepared modules which can be integrated into programs written in all popular engineering programming languages. These libraries not only contain excellent routines for solving simultaneous linear equations and the eigenvalues of characteristic matrices, but also embody procedures guarding against ill-conditioned matrices which lead to large computational errors. Special matrix techniques called relaxation are used to solve partial differential equations on the computer. A typical problem requires setting up a grid of hundreds or thousands of points to describe the region and expressing the equation at each point by finite-difference methods. The resulting matrix is very sparse with a regular pattern of nonzero elements. The form of the matrix circumvents the need for handling large arrays of numbers in the computer and avoids problems in computational accuracy normally found in dealing with extremely large matrices. The computer is an excellent tool for handling optimization problems. Mathematically these problems are formulated as problems in finding the maximum or minimum of a nonlinear equation. The excellent techniques that have been developed can deal effectively with the unique complexities these problems have, such as saddle points which represent both a maximum and a minimum. Another class of problems, called linear programming problems, is characterized by the linear constraint of many variables which plot into regions outlined by multidimensional planes (in the two-dimensional case, the region is a plane enclosed by straight lines). Techniques have been developed to find the optimal solution of the variables satisfying some given value or cost objective function. The solution to the problem proceeds by searching the corners of the region defined by the constraining equations to find points which represent minimum points of a cost function or maximum points of a value function.

SOFTWARE SYSTEMS

The best known and most widely used techniques for solving statistical problems are those of linear statistics. These involve the techniques of least squares (otherwise known as regression). For some problems these techniques do not suffice, and more specialized techniques involving nonlinear statistics must be used, albeit a solution may not exist. Artificial intelligence (AI) is the study and implementation of programs that model knowledge systems and exhibit aspects of intelligence in problem solving. Typical areas of application are in learning, linguistics, pattern recognition, decision making, and theorem proving. In AI, the computer serves to search a collection of heuristic rules to find a match with a current situation and to make inferences or otherwise reorganize knowledge into more useful forms. AI techniques have been utilized to build sophisticated systems, called expert systems, to aid in producing a timely response in problems involving a large number of complex conditions. Operating Systems

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Multiprogramming operating systems process several jobs concurrently. A job may be initiated any time memory and other resources which it needs become available. Many jobs may be simultaneously active in the system and maintained in a partial state of completion. The order of execution depends on the priority assignments. Jobs are executed to completion or put into a wait state until a pending request for service has been satisfied. It should be noted that, while the CPU can execute only a single program at any moment of time, operations with peripheral and storage devices can occur concurrently. Timesharing operating systems process jobs in a way similar to multiprogramming except for the added feature that each job is given a short slice of the available time to complete its tasks. If the job has not been completed within its time slice or if it requests a service from an external device, it is put into a wait status and control passes to the next job. Effectively, the length of the time slice determines the priority of the job. Program Preparation Facilities

The operating system provides the services that support the needs that computer programs have in common during execution. Any list of services would include those needed to configure the resources that will be made available to the users, to attach hardware units (e.g., memory modules, storage devices, coprocessors, and peripheral devices) to the existing configuration, to detach modules, to assign default parameters to the hardware and software units, to set up and schedule users’ tasks so as to resolve conflicts and optimize throughput, to control system input and output devices, to protect the system and users’ programs from themselves and from each other, to manage storage space in the file devices, to protect file devices from faults and illegal use, to account for the use of the system, and to handle in an orderly way any exception which might be encountered during program execution. A well-designed operating system provides these services in a user-friendly environment and yet makes itself and the computer operating staff transparent to the user. The design of a computer operating system depends on the number of users which can be expected. The focus of single-user systems relies on the monitor to provide a user-friendly system through dialog menus with icons, mouse operations, and templets. Table 2.2.6 lists some popular operating systems for PCs by their trademark names. The design of a multiuser system attempts to give each user the impression that he/she is the lone user of the system. In addition to providing the accoutrements of a user-friendly system, the design focuses on the order of processing the jobs in an attempt to treat each user in a fair and equitable fashion. The basic issues for determining the order of processing center on the selection of job queues: the number of queues (a simple queue or a mix of queues), the method used in scheduling the jobs in the queue (first come–first served, shortest job next, or explicit priorities), and the internal handling of the jobs in the queue (batch, multiprogramming, or timesharing). Table 2.2.6 Some Popular PC Operating Systems Trademark

Supplier

Windows Unix Sun/OS Macintosh Linux

Microsoft Corp. Unix Systems Laboratory Inc. Sun Microsystems Inc. Apple Computer Inc. Multiple suppliers

Batch operating systems process jobs in a sequential order. Jobs are collected in batches and entered into the computer with individual job instructions which the operating system interprets to set up the job, to allocate resources needed, to process the job, and to provide the input/ output. The operating system processes each job to completion in the order it appears in the batch. In the event a malfunction or fault occurs during execution, the operating system terminates the job currently being executed in an orderly fashion before initiating the next job in sequence.

For the user, the crucial part of a language system is the grammar which specifies the language syntax and semantics that give the symbols and rules used to compose acceptable statements and the meaning associated with the statements. Compared to natural languages, computer languages are more precise, have a simpler structure, and have a clearer syntax and semantics that allows no ambiguities in what one writes or what one means. For a program to be executed, it must eventually be translated into a sequence of basic machine instructions. The statements written by a user must first be put on some machinereadable medium or typed on a keyboard for entry into the machine. The translator (compiler) program accepts these statements as input and translates (compiles) them into a sequence of basic machine instructions which form the executable version of the program. After that, the translated (compiled) program can be run. During the execution of a program, a run-time program must also be present in the memory. The purpose of the run-time system is to perform services that the user’s program may require. For example, in case of a program fault, the run-time system will identify the error and terminate the program in an orderly manner. Some language systems do not have a separate compiler to produce machine-executable instructions. Instead the run-time system interprets the statements as written, converts them into a pseudo-code, and executes the coded version. Commonly needed functions are made available as prepared modules, either as an integral part of the language or from stored libraries. The documentation of these functions must be studied carefully to assure correct selection and utilization. Languages may be classified as procedure-oriented or problemoriented. With procedure-oriented languages, all the detailed steps must be specified by the user. These languages are usually characterized as being more verbose than problem-oriented languages, but are more flexible and can deal with a wider range of problems. Problem-oriented languages deal with more specialized classes of problems. The elements of problem-oriented languages are usually familiar to a knowledgeable professional and so are easier to learn and use than procedure-oriented languages. The most elementary form of a procedure-oriented language is called an assembler. This class of language permits a computer program to be written directly in basic computer instructions using mnemonic operators and symbolic operands. The assembler’s translator converts these instructions into machine-usable form. A further refinement of an assembler permits the use of macros. A macro identifies, by an assigned name and a list of formal parameters, a sequence of computer instructions written in the assembler’s format and stored in its subroutine library. The macroassembler includes these macro instructions in the translated program along with the instructions written by the programmer. Besides these basic language systems there exists a large variety of other language systems. These are called higher-level language systems since they permit more complex statements than are permitted by a

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COMPUTERS

macroassembler. They can also be used on machines produced by different manufacturers or on machines with different instruction repertoires. One language of historical value is ALGOL 60. It is a landmark in the theoretical development of computer languages. It was designed and standardized by an international committee whose goal was to formulate a language suitable for publishing computer algorithms. Its importance lies in the many language features it introduced which are now common in the more recent languages which succeeded it and in the scientific notation which was used to define it. FORTRAN (FORmula TRANslator) was one of the first languages catering to the engineering and scientific community where algebraic formulas specify the computations used within the program. It has been standardized several times. Each version has expanded the language features and has removed undesirable features which lead to unstructured programs. The PASCAL language couples the ideas of ALGOL 60 to those of structured programming. By allowing only appropriate statement types, it guarantees that any program written in the language will be well structured. In addition, the language introduced new data types and allows programmers to define new complex data structures based on the primitive data types. The definition of the Ada language was sponsored by the Department of Defense as an all-encompassing language for the development and maintenance of very large, software-intensive projects over their life cycle. While it meets software engineering objectives in a manner similar to Pascal, it has many other features not normally found in programming languages. Like other attempts to formulate very large all-inclusive languages, it is difficult to learn and has not found popular favor. Nevertheless, its many unique features make it especially valuable in implementing programs which cannot be easily implemented in other languages (e.g., programs for parallel computations in embedded computers). By edict, subsets of Ada were forbidden. Modula-2 was designed to retain the inherent simplicity of PASCAL but include many of the advanced features of Ada. Its advantage lies in implementing large projects involving many programmers. The compilers for this language have rigorous interface cross-checking mechanisms to avoid poor interfaces between components. Another troublesome area is in the implicit use of global data. Modula-2 retains the Ada facilities that allow programmers to share data and avoids incorrectly modifying the data in different program units. The C language was developed by AT&T’s Bell Laboratories and subsequently standardized by ANSI. It has a reputation for translating programs into compact and fast code, and for allowing program segments to be precompiled. Its strength rests in the flexibility of the language; for example, it permits statements from other languages to be included in-line in a C program and it offers the largest selection of operators that mirror those available in an assembly language. Because of its flexibility, programs written in C can become unreadable. The C language was also developed by AT&T Bell Laboratories and standardized by ANSI. The C language was designed as an extension of C so that the concepts required to design and implement a program would be expressed more directly and concisely. Use of classes to embody the design concept is one of the primary concepts of C. Problem-oriented languages have been developed for every discipline. A language might deal with a specialized application within an engineering field, or it might deal with a whole gamut of applications covering one or more fields. A class of problem-oriented languages that deserves special mention are those for solving problems in discrete simulation. GPSS, Simscript, and SIMULA are among the most popular. A simulation (another word for model) of a system is used whenever it is desirable to watch a succession of many interrelated events or when there is interplay between the system under study and outside forces. Examples are problems in human-machine interaction and in the modeling of business systems. Typical human-machine problems are the servicing of automatic equipment by a crew of operators (to study crew size and assignments, typically), or responses by shared maintenance crews to equipment subject to unpredictable (random) breakdown. Business models often involve

transportation and warehousing studies. A business model could also study the interactions between a business and the rest of the economy such as competitive buying in a raw materials market or competitive marketing of products by manufacturers. Physical or chemical systems may also be modeled. For example, to study the application of automatic control values in pipelines, the computer model consists of the control system, the valves, the piping system, and the fluid properties. Such a model, when tested, can indicate whether fluid hammer will occur or whether valve action is fast enough. It can also be used to predict pressure and temperature conditions in the fluid when subject to the valve actions. Another class of problem-oriented languages makes the computer directly accessible to the specialist with little additional training. This is achieved by permitting the user to describe problems to the computer in terms that are familiar in the discipline of the problem and for which the language is designed. Two approaches are used. Figures 2.2.9 and 2.2.10 illustrate these. One approach sets up the computer program directly from the mathematical equations. In fact, problems were formulated in this manner in the past, where analog computers were especially well-suited. Anyone familiar with analog computers finds the transitions to these languages easy. Figure 2.2.9 illustrates this approach using the MIMIC language to write the program for the solution of the initial-value problem: $ # # M y 1 Z y 1 Ky 5 1 and y s0d 5 ys0d 5 0 MIMIC is a digital simulation language used to solve systems of ordinary differential equations. The key step in setting up the solution is to isolate the highest-order derivative on the left-hand side of the equation and equate it to an expression composed of the remaining terms. For the equation above, this results in: # $ y 5 s1 2 Z y 2 Kyd/M The highest-order derivative is derived by equating it to the expression on the right-hand side of the equation. The lower-order derivatives in the expression are generated successively by integrating the highestorder derivative. The MIMIC language permits the user to write these statements in a format closely resembling mathematical notation. The alternate approach used in problem-oriented languages permits the setup to be described to the computer directly from the block diagram of the physical system. Figure 2.2.10 illustrates this approach using the SCEPTRE language. SCEPTRE statements are written under headings and subheadings which identify the type of component being described. This language may be applied to network problems of electrical digital-logic elements, mechanical-translation or rotational elements, or transfer-function blocks. The translator for this language develops and sets up the equations directly from this description of the

MIMIC statements

Explanation

DY2  (1Z * DY1  K * Y)/M

Differential equation to be solved. “*” is used for multiplication and DY2, DY1, and Y $ are # defined mnemonics for y , y , and y. INT (A, B) is used to perform integration. It forms successive values of B   Adt. T is a reserved name representing the independent variable. This statement will terminate execution when T $ 10. Values must be furnished for M, K, and Z. An input with these values must appear after the END card. Three point plots $ # are produced on the line printer; y , y , and y vs. t.

DY1  INT(DY2,0.) Y  INT(DY1,0.) FIN(T,10.)

CON(M, K, Z)

PLO(T,DY2) PLO(T,DY1) PLO(T,Y) END

Necessary last statement.

Fig. 2.2.9 Illustration of a MIMIC program.

SOFTWARE SYSTEMS A node is assigned to: • ground • any mass • point between two elements The prefix of element name specifies its type, i.e., M for mass, K for spring, D for damper, and R for force. (a) SCEPTRE statements MECHANICAL DESCRIPTION ELEMENTS M1, 1  3  10. K1, 1  2  40. D1, 2  3  .5 R1, 1  3  7.32 OUTPUT SM1, VM1

RUN CONTROL STOPTIME  10. END

Explanation

Specifies the elements and their position in the diagram using the node numbers.

Results are listed on the line printer. Prefix on the element specifies the quantity to be listed; S for displacement, V for velocity. TIME is reserved name for independent variable. Statement will terminate execution of program when TIME is equal to or greater than 10. Necessary statement. (b)

Fig. 2.2.10 Illustration of SCEPTRE program. (a) Problem to be solved; (b) SCEPTRE program.

network diagram, and so relieves the user from the mathematical aspects of the problem. Application Packages

An application package differs from a language in that its components have been organized to solve problems in a particular application rather than to create the components themselves. The user interacts with the package by initiating the operations and providing the data. From an operational view, packages are built to minimize or simplify interactions with the users by using a menu to initiate operations and entering the data through templets. Perhaps the most widely used application package is the word processor. The objective of a word processor is to allow users to compose text in an electronically stored format which can be corrected or modified, and from which a hard copy can be produced on demand. Besides the basic typewriter operations, it contains functions to manipulate text in blocks or columns, to create headers and footers, to number pages, to find and correct words, to format the data in a variety of ways, to create labels, and to merge blocks of text together. The better word processors have an integrated dictionary, a spelling checker to find and correct misspelled words, a grammar checker to find grammatical errors, and a thesaurus. They often have facilities to prepare complex mathematical equations and to include and manipulate graphical artwork, including editing color pictures. When enough page- and document formatting capability has been added, the programs are known as desktop publishing programs. One of the programs that contributed to the early acceptance of personal computers was the spread sheet program. These programs simulate the common spread sheet with its columns and rows of interrelated data. The computerized approach has the advantage that the equations are stored so that the results of a change in data can be shown quickly after any change is made in the data. Modern spread sheet programs have many capabilities, including the ability to obtain information from other spread sheets, to produce a variety of reports, and to prepare equations which have complicated logical aspects. Tools for project management have been organized into commercially available application packages. The objectives of these programs

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are in the planning, scheduling, and controlling the time-oriented activities describing the projects. There are two basically similar techniques used in these packages. One, called CPM (critical path method), assumes that the project activities can be estimated deterministically. The other, called PERT (project evaluation and review technology), assumes that the activities can be estimated probabilistically. Both take into account such items as the requirement that certain tasks cannot start before the completion of other tasks. The concepts of critical path and float are crucial, especially in scheduling the large projects that these programs are used for. In both cases tools are included for estimating project schedules, estimating resources needed and their schedules, and representing the project activities in graphical as well as tabular form. A major use of the digital computer is in data reduction, data analysis, and visualization of data. In installations where large amounts of data are recorded and kept, it is often advisable to reduce the amount of data by ganging the data together, by averaging the data with numerical filters to reduce the amount of noise, or by converting the data to a more appropriate form for storage, analysis, visualization, or future processing. This application has been expanded to produce systems for evaluation, automatic testing, and fault diagnosis by coupling the data acquisition equipment to special peripherals that automatically measure and record the data in a digital format and report the data as meaningful, nonphysically measurable parameters associated with a mathematical model. Computer-aided design/computer-aided manufacturing (CAD/CAM) is an integrated collection of software tools which have been designed to make way for innovative methods of fabricating customized products to meet customer demands. The goal of modern manufacturing is to process orders placed for different products sooner and faster, and to fabricate them without retooling. CAD has the tools for prototyping a design and setting up the factory for production. Working within a framework of agile manufacturing facilities that features automated vehicles, handling robots, assembly robots, and welding and painting robots, the factory sets itself up for production under computer control. Production starts with the receipt of an order on which customers may pick options such as color, size, shapes, and features. Manufacturing proceeds with greater flexibility, quality, and efficiency in producing an increased number of products with a reduced workforce. Effectively, CAD/CAM provides for the ultimate just-in-time (JIT) manufacturing. Collaboration software is a set of software tools that have been designed to enhance timely and effective communication within a software preparation group. The use of collaboration software has been an important tool to improve the speed and effectiveness of communication for local teams and especially for geographically dispersed teams. With such software, all team members have access to current information such as financial or experimental data, reports, and presentations. Proposals and updates can be provided rapidly to all team members in response to changing conditions. Working within the framework of the collaboration tools can also provide the team with a history that can be very valuable in addressing new problems and in providing new team members with information required to allow them to quickly become effective members of the team. Two other types of application package illustrate the versatility of data management techniques. One type ties on-line equipment to a computer for collecting real-time data from the production lines. An animated, pictorial display of the production lines forms the heart of the system, allowing supervision in a central control station to continuously track operations. The other type collects time-series data from the various activities in an enterprise. It assists in what is known as management by exception. It is especially useful where the detailed data is so voluminous that it is feasible to examine it only in summaries. The data elements are processed and stored in various levels of detail in a seamless fashion. The system stores the reduced data and connects it to the detailed data from which it was derived. The application package allows management, through simple computer operations, to detect a problem at a higher level and to locate and pinpoint its cause through examination of successively lower levels.

Section

3

Mechanics of Solids and Fluids BY

PETER L. TEA, JR. Professor of Physics Emeritus, The City College of The City University of

New York VITTORIO (RINO) CASTELLI Senior Research Fellow, Retired, Xerox Corp.; Engineering

Consultant J. W. MURDOCK Late Consulting Engineer LEONARD MEIROVITCH University Distinguished Professor Emeritus, Department of

Engineering Science and Mechanics, Virginia Polytechnic Institute and State University

3.1 MECHANICS OF SOLIDS by Peter L. Tea, Jr. Physical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 Systems and Units of Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 Statics of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Center of Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Dynamics of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14 Work and Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17 Impulse and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-18 Gyroscopic Motion and the Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19 3.2 FRICTION by Vittorio (Rino) Castelli Static and Kinetic Coefficients of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 3-20 Rolling Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25 Friction of Machine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25 3.3 MECHANICS OF FLUIDS by J. W. Murdock Fluids and Other Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-31 Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33

Fluid Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-36 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-37 Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-41 Dynamic Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-43 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-44 Forces of Immersed Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-46 Flow in Pipes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-47 Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-50 ASME Pipeline Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53 Pitot Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-57 ASME Weirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-57 Open-Channel Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59 Flow of Liquids from Tank Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60 Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-61 Computational Fluid Dynamics (CFD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-61 3.4 VIBRATION by Leonard Meirovitch Single-Degree-of-Freedom Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-61 Multi-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-70 Distributed-Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-72 Approximate Methods for Distributed Systems . . . . . . . . . . . . . . . . . . . . . . 3-75 Vibration-Measuring Instruments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-78

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3.1 MECHANICS OF SOLIDS by Peter L. Tea, Jr. REFERENCES: Beer and Johnston, “Mechanics for Engineers,” McGraw-Hill. Ginsberg and Genin, “Statics and Dynamics,” Wiley. Higdon and Stiles, “Engineering Mechanics,” Prentice-Hall. Holowenko, “Dynamics of Machinery,” Wiley. Housnor and Hudson, “Applied Mechanics,” Van Nostrand. Meriam, “Statics and Dynamics,” Wiley. Mabie and Ocvirk, “Mechanisms and Dynamics of Machinery,” Wiley. Synge and Griffith, “Principles of Mechanics,” McGraw-Hill. Timoshenko and Young, “Advanced Dynamics,” McGraw-Hill. Timoshenko and Young, “Engineering Mechanics,” McGraw-Hill.

PHYSICAL MECHANICS Definitions Force is the action of one body on another which will cause acceleration

of the second body unless acted on by an equal and opposite action counteracting the effect of the first body. It is a vector quantity. Time is a measure of the sequence of events. In newtonian mechanics it is an absolute quantity. In relativistic mechanics it is relative to the frames of reference in which the sequence of events is observed. The common unit of time is the second. Inertia is that property of matter which causes a resistance to any change in the motion of a body. Mass is a quantitative measure of inertia. Acceleration of Gravity Every object which falls in a vacuum at a given position on the earth’s surface will have the same acceleration g. Accurate values of the acceleration of gravity as measured relative to the earth’s surface include the effect of the earth’s rotation and flattening at the poles. The international gravity formula for the acceleration of gravity at the earth’s surface is g  32.0881(1  0.005288 sin2   0.0000059 sin2 2) ft/s2, where  is latitude in degrees. For extreme accuracy, the local acceleration of gravity must also be corrected for the presence of large water or land masses and for height above sea level. The absolute acceleration of gravity for a nonrotating earth discounts the effect of the earth’s rotation and is rarely used, except outside the earth’s atmosphere. If g0 represents the absolute acceleration at sea level, the absolute value at an altitude h is g  g0R2/(R  h)2, where R is the radius of the earth, approximately 3,960 mi (6,373 km). Weight is the resultant force of attraction on the mass of a body due to a gravitational field. On the earth, units of weight are based upon an acceleration of gravity of 32.1740 ft/s2 (9.80665 m/s2). Linear momentum is the product of mass and the linear velocity of a particle and is a vector. The moment of the linear-momentum vector about a fixed axis is the angular momentum of the particle about that fixed axis. For a rigid body rotating about a fixed axis, angular momentum is defined as the product of moment of inertia and angular velocity, each measured about the fixed axis. An increment of work is defined as the product of an incremental displacement and the component of the force vector in the direction of the displacement or the component of the displacement vector in the direction of the force. The increment of work done by a couple acting on a body during a rotation of d in the plane of the couple is dU  M d. Energy is defined as the capacity of a body to do work by reason of its motion or configuration (see Work and Energy). A vector is a directed line segment that has both magnitude and direction. In script or text, a vector is distinguished from a scalar V by a boldface-type V. The magnitude of the scalar is the magnitude of the vector, V  |V|. A frame of reference is a specified set of geometric conditions to which other locations, motion, and time are referred. In newtonian mechanics, the fixed stars are referred to as the primary (inertial) frame of reference. Relativistic mechanics denies the existence of a primary 3-2

reference frame and holds that all reference frames must be described relative to each other. SYSTEMS AND UNITS OF MEASUREMENTS

In absolute systems, the units of length, mass, and time are considered fundamental quantities, and all other units including that of force are derived. In gravitational systems, the units of length, force, and time are considered fundamental qualities, and all other units including that of mass are derived. In the SI system of units, the unit of mass is the kilogram (kg) and the unit of length is the metre (m). A force of one newton (N) is derived as the force that will give 1 kilogram an acceleration of 1 m/s2. In the English engineering system of units, the unit of mass is the pound mass (lbm) and the unit of length is the foot (ft). A force of one pound (1 lbf) is the force that gives a pound mass (1 lbm) an acceleration equal to the standard acceleration of gravity on the earth, 32.1740 ft/s2 (9.80665 m/s2). A slug is the mass that will be accelerated 1 ft/s2 by a force of 1 lbf. Therefore, 1 slug  32.1740 lbm. When described in the gravitational system, mass is a derived unit, being the constant of proportionality between force and acceleration, as determined by Newton’s second law. General Laws

NEWTON’S LAWS I. If a balanced force system acts on a particle at rest, it will remain at rest. If a balanced force system acts on a particle in motion, it will remain in motion in a straight line without acceleration. II. If an unbalanced force system acts on a particle, it will accelerate in proportion to the magnitude and in the direction of the resultant force. III. When two particles exert forces on each other, these forces are equal in magnitude, opposite in direction, and collinear. Fundamental Equation The basic relation between mass, acceleration, and force is contained in Newton’s second law of motion. As applied to a particle of mass, F  ma, force  mass  acceleration. This equation is a vector equation, since the direction of F must be the direction of a, as well as having F equal in magnitude to ma. An alternative form of Newton’s second law states that the resultant force is equal to the time rate of change of momentum, F  d(mv)/dt. Law of the Conservation of Mass The mass of a body remains unchanged by any ordinary physical or chemical change to which it may be subjected. (True in classical mechanics.) Law of the Conservation of Energy The principle of conservation of energy requires that the total mechanical energy of a system remain unchanged if it is subjected only to forces which depend on position or configuration. Law of the Conservation of Momentum The linear momentum of a system of bodies is unchanged if there is no resultant external force on the system. The angular momentum of a system of bodies about a fixed axis is unchanged if there is no resultant external moment about this axis. Law of Mutual Attraction (Gravitation) Two particles attract each other with a force F proportional to their masses m1 and m2 and inversely proportional to the square of the distance r between them, or F  Gm1m2/r2, in which G is the gravitational constant. The value of the gravitational constant is G  6.673  1011 m3/kg  s2 in SI or absolute units, or G  3.44  108 lbf  ft2/slug2 in engineering gravitational units. It should be pointed out that the unit of force F in the SI system is the newton and is derived, while the unit force in the gravitational system is the pound-force and is a fundamental quantity.

STATICS OF RIGID BODIES EXAMPLE. Each of two solid steel spheres 6 in in diam will weigh 32.0 lb on the earth’s surface. This is the force of attraction between the earth and the steel sphere. The force of mutual attraction between the spheres if they are just touching is 0.000000136 lbf. (For spheres, center-to-center distance may be used.)

STATICS OF RIGID BODIES General Considerations

If the forces acting on a rigid body do not produce any acceleration, they must neutralize each other, i.e., form a system of forces in equilibrium. Equilibrium is said to be stable when the body with the forces acting upon it returns to its original position after being displaced a very small amount from that position; unstable when the body tends to move still farther from its original position than the very small displacement; and neutral when the forces retain their equilibrium when the body is in its new position. External and Internal Forces The forces by which the individual particles of a body act on each other are known as internal forces. All other forces are called external forces. If a body is supported by other bodies while subject to the action of forces, deformations and forces will be produced at the points of support or contact and these internal forces will be distributed throughout the body until equilibrium exists and the body is said to be in a state of tension, compression, or shear. The forces exerted by the body on the supports are known as reactions. They are equal in magnitude and opposite in direction to the forces with which the supports act on the body, known as supporting forces. The supporting forces are external forces applied to the body. In considering a body at a definite section, it will be found that all the internal forces act in pairs, the two forces being equal and opposite. The external forces act singly. General Law When a body is at rest, the forces acting externally to it must form an equilibrium system. This law will hold for any part of the body, in which case the forces acting at any section of the body become external forces when the part on either side of the section is considered alone. In the case of a rigid body, any two forces of the same magnitude, but acting in opposite directions in any straight line, may be added or removed without change in the action of the forces acting on the body, provided the strength of the body is not affected.

3-3

Resultant of Any Number of Forces Applied to a Rigid Body at the Same Point Resolve each of the given forces F into components

along three rectangular coordinate axes. If A, B, and C are the angles made with XX, YY, and ZZ, respectively, by any force F, the components will be F cos A along XX, F cos B along YY, F cos C along ZZ; add the components of all the forces along each axis algebraically and obtain F cos A  X along XX, F cos B  Y along YY, and F cos C  Z along ZZ. The resultant R 5 2sXd2 1 sYd2 1 sZd2. The angles made by the resultant with the three axes are Ar with XX, Br with YY, Cr with ZZ, where cos Ar 5 X/R

cos Br 5 Y/R

cos Cr 5 Z/R

The direction of the resultant can be determined by plotting the algebraic sums of the components. If the forces are all in the same plane, the components of each of the forces along one of the three axes (say ZZ) will be 0; i.e., angle Cr  90 and R 5 2sXd2 1 sYd2, cos Ar  X/R, and cos Br  Y/R. For equilibrium, it is necessary that R  0; i.e., X, Y, and Z must each be equal to zero. General Law In order that a number of forces acting at the same point shall be in equilibrium, the algebraic sum of their components along any three coordinate axes must each be equal to zero. When the forces all act in the same plane, the algebraic sum of their components along any two coordinate axes must each equal zero. When the Forces Form a System in Equilibrium Three unknown forces can be determined if the lines of action of the forces are all known and are in different planes. If the forces are all in the same plane, the lines of action being known, only two unknown forces can be determined. If the lines of action of the unknown forces are not known, only one unknown force can be determined in either case. Couples and Moments Couple Two parallel forces of equal magnitude (Fig. 3.1.3) which act in opposite directions and are not collinear form a couple. A couple cannot be reduced to a single force.

Composition, Resolution, and Equilibrium of Forces

The resultant of several forces acting at a point is a force which will produce the same effect as all the individual forces acting together. Forces Acting on a Body at the Same Point The resultant R of two forces F1 and F2 applied to a rigid body at the same point is represented in magnitude and direction by the diagonal of the parallelogram formed by F1 and F2 (see Figs. 3.1.1 and 3.1.2). R 5 2F 21 1 F 22 1 2F1F2 cos a sin a1 5 sF2 sin ad/R

sin a2 5 sF1 sin ad/R

When a 5 908, R 5 2F 21 1 F 22, sin a1 5 F2/R, and sin a2 5 F1/R. When a 5 08, R 5 F1 1 F2 r When a 5 1808, R 5 F1 2 F2

Forces act in same straight line.

A force R may be resolved into two component forces intersecting anywhere on R and acting in the same plane as R, by the reverse of the operation shown by Figs. 3.1.1 and 3.1.2; and by repeating the operation with the components, R may be resolved into any number of component forces intersecting R at the same point and in the same plane.

Fig. 3.1.1

Fig. 3.1.2

Fig. 3.1.3 Displacement and Change of a Couple The forces forming a couple may be moved about and their magnitude and direction changed, provided they always remain parallel to each other and remain in either the original plane or one parallel to it, and provided the product of one of the forces and the perpendicular distance between the two is constant and the direction of rotation remains the same. Moment of a Couple The moment of a couple is the product of the magnitude of one of the forces and the perpendicular distance between the lines of action of the forces. Fa  moment of couple; a  arm of couple. If the forces are measured in pounds and the distance a in feet, the unit of rotation moment is the foot-pound. If the force is measured in newtons and the distance in metres, the unit is the newton-metre. In the cgs system the unit of rotation moment is 1 cm-dyne. Rotation moments of couples acting in the same plane are conventionally considered to be positive for counterclockwise moments and negative for clockwise moments, although it is only necessary to be consistent within a given problem. The magnitude, direction, and sense of rotation of a couple are completely determined by its moment axis, or moment vector, which is a line drawn perpendicular to the plane in which the couple acts, with an arrow indicating the direction from which the couple will appear to have right-handed rotation; the length of the line represents the magnitude of the moment of the couple. Figure 3.1.4a shows a counterclockwise couple, designated . Figure 3.1.4b shows a clockwise couple, designated .

3-4

MECHANICS OF SOLIDS

T

a single vector (having no specific line of action) perpendicular to the plane of the couple. If you let the relaxed fingers of your right hand represent the “swirl” of the forces in the couple, your stiff right thumb will indicate the direction of the vector. Vector couples may be added vectorially in the same manner by which concurrent forces are added. Couples lying in the same or parallel planes are added algebraically. Let 28 lbf  ft (38 N  m), 42 lbf  ft (57 N  m), and 70 lbf  ft (95 N  m) be the moments of three couples in the same or parallel planes; their resultant is a single couple lying in the same or in a parallel plane, whose moment is M  28  42  70  56 lbf  ft (M  38  57  95  76 N  m). Moment of couple = −180 lbf . ft 18 lbf from couple Y F

F

F

F = 18 lbf Point p

x

F X

(+)

(−)

(a)

(b)

Fig. 3.1.4

X 10 ft Y

18 lbf from couple

Fig. 3.1.5

If the polygon formed by the moment vectors of several couples closes itself, the couples form an equilibrium system. Two couples will balance each other

when they lie in the same or parallel planes and have the same moment in magnitude, but opposite in sign. Combination of a Couple and a Single Force in the Same Plane

(Fig. 3.1.5) Given a force F  18 lbf (80 N) acting as shown at distance x from YY, and a couple whose moment is 180 lbf  ft (244 N  m) in the same or parallel plane, to find the resultant. A couple may be changed to any other couple in the same or a parallel plane having the same moment and same sign. Let the couple consist of two forces of 18 lbf (80 N) each and let the arm be 10 ft (3.05 m). Place the couple in such a manner that one of its forces is opposed to the given force at p. This force of the couple and the given force being of the same magnitude and opposite in direction will neutralize each other, leaving the other force of the couple acting at a distance of 10 ft (3.05 m) from p and parallel and equal to the given force 18 lbf (80 N). General Rule The resultant of a couple and a single force lying in the same or parallel planes is a single force, equal in magnitude, in the same direction and parallel to the single force, and acting at a distance from the line of action of the single force equal to the moment of the couple divided by the single force. The moment of the resultant force about any point on the line of action of the given single force must be of the same sense as that of the couple, positive if the moment of the couple is positive, and negative if the moment of the couple is negative. If the moment of the couple in Fig. 3.1.5 had been  instead of , the resultant would have been a force of 18 lbf (80 N) acting in the same direction and parallel to F, but at a distance of 10 ft (3.05 m) to the right of it, making the moment of the resultant about any point on F positive. To effect a parallel displacement of a single force F over a distance a, a couple whose moment is Fa must be added to the system. The sense of the couple will depend upon which way it is desired to displace force F. The moment of a force with respect to a point is the product of the force F and the perpendicular distance from the point to the line of action of the force. The Moment of a Force with Respect to a Straight Line If the force is resolved into components parallel and perpendicular to the given line, the moment of the force with respect to the line is the product of the magnitude of the perpendicular component and the distance from its line of action to the given line.

of action makes angles Ar , Br , and Cr with axes XX, YY, and ZZ, where cos Ar  X/R, cos Br  Y/R, and cos Cr  Z/R; and there are three couples which may be combined by their moment vectors into a single resultant couple having the moment M r 5 2sM xd2 1 sM yd2 1 sM zd2, whose moment vector makes angles of Am , Bm , and Cm with axes XX, YY, and ZZ, such that cos Am  Mx /Mr , cos Bm  My /Mr, cos Cm  Mz /Mr. If this single resulting couple is in the same plane as the single resulting force at the origin or a plane parallel to it, the system may be reduced to a single force R acting at a distance from R equal to Mr /R. If the couple and force are not in the same or parallel planes, it is impossible to reduce the system to a single force. If R  0, i.e., if X, Y, and Z all equal zero, the system will reduce to a single couple whose moment is Mr. If Mr  0, i.e., if Mx, My, and Mz all equal zero, the resultant will be a single force R. When the forces are all in the same plane, the cosine of one of the angles Ar, Br, or Cr  0, say, Cr  90. Then R 5 2sXd2 1 sYd2, M r 5 2M 2x 1 M 2y , and the final resultant is a force equal and parallel to R, acting at a distance from R equal to Mr /R. A system of forces in the same plane can always be replaced by either a couple or a single force. If R  0 and Mr  0, the resultant is a couple. If Mr  0 and R  0, the resultant is a single force. A rigid body is in equilibrium when acted upon by a system of forces whenever R  0 and Mr  0, i.e., when the following six conditions hold true: X  0, Y  0, Z  0, Mx  0, My  0, and Mz  0. When the system of forces is in the same plane, equilibrium prevails when the following three conditions hold true: X  0, Y  0, M  0. Forces Applied to Support Rigid Bodies

The external forces in equilibrium acting upon a body may be statically determinate or indeterminate according to the number of unknown forces existing. When the forces are all in the same plane and act at a common point, two unknown forces may be determined if their lines of action are known, or one unknown force if its line of action is unknown. When the forces are all in the same plane and are parallel, two unknown forces may be determined if the lines of action are known, or one unknown force if its line of action is unknown. When the forces are anywhere in the same plane, three unknown forces may be determined if their lines of action are known, if they are not parallel or do not pass through a common point; if the lines of action are unknown, only one unknown force can be determined. If the forces all act at a common point but are in different planes, three unknown forces can be determined if the lines of action are known, one if unknown. If the forces act in different planes but are parallel, three unknown forces can be determined if their lines of action are known, one if unknown. The first step in the solution of problems in statics is the determination of the supporting forces. The following data are required for the complete knowledge of supporting forces: magnitude, direction, and point of application. According to the nature of the problem, none, one, or two of these quantities are known. One Fixed Support The point of application, direction, and magnitude of the load are known. See Fig. 3.1.6. As the body on which the forces act is in equilibrium, the supporting force P must be equal in magnitude and opposite in direction to the resultant of the loads L. In the case of a rolling surface, the point of application of the support is obtained from the center of the connecting bolt A (Fig. 3.1.7), both the direction and magnitude being unknown. The point of application and

Forces with Different Points of Application Composition of Forces If each force F is resolved into components parallel to three rectangular coordinate axes XX, YY, ZZ, the magnitude of the resultant is R 5 2sXd2 1 sYd2 1 sZd2, and its line

Fig. 3.1.6

Fig. 3.1.7

STATICS OF RIGID BODIES

line of action of the support at B are known, being determined by the rollers. When three forces acting in the same plane on the same rigid body are in equilibrium, their lines of action must pass through the same point O. The load L is known in magnitude and direction. The line of action of the support at B is known on account of the rollers. The point of application of the support at A is known. The three forces are in equilibrium and are in the same plane; therefore, the lines of action must meet at the point O. In the case of the rolling surfaces shown in Fig. 3.1.8, the direction of the support at A is known, the magnitude unknown. The line of action and point of application of the supporting force at B are known, its

3-5

closing sides of the individual triangles, the magnitude and direction of the resultant R of any number of forces in the same plane and intersecting

Fig. 3.1.10

at a single point can be found. In Fig. 3.1.11 the lines representing the forces start from point O, and in the force polygon (Fig. 3.1.12) they are joined in any order, the arrows showing their directions following around the polygon in the same direction. The magnitude of the resultant at the point of application of the forces is represented by the closing side R of the force polygon; its direction, as shown by the arrow, is counter to that in the other sides of the polygon. If the forces are in equilibrium, R must equal zero, i.e., the force polygon must close.

Fig. 3.1.8

Fig. 3.1.9

magnitude unknown. The lines of action of the three forces must meet in a point, and the supporting force at A must be perpendicular to the plane XX. If a member of a truss or frame in equilibrium is pinned at two points and loaded at these two points only, the line of action of the forces exerted on the member or by the member at these two points must be along a line connecting the pins. If the external forces acting upon a rigid body in equilibrium are all in the same plane, the equations X  0, Y  0, and M  0 must be satisfied. When trusses, frames, and other structures are under discussion, these equations are usually used as V  0, H  0, M  0, where V and H represent vertical and horizontal components, respectively. In Fig. 3.1.9, the directions and points of application of the supporting forces are known, but all three are of unknown magnitudes. Including load L, there are four forces, and the three unknown magnitudes may be determined by X 5 0, Y 5 0, and M 5 0. The supports are said to be statically determinate when the laws of equilibrium are sufficient for their determination. When the conditions are not sufficient for the determination of the supports or other forces, the structure is said to be statically indeterminate; the unknown forces can then be determined from considerations involving the deformation of the material. When several bodies are so connected to one another as to make up a rigid structure, the forces at the points of connection must be considered as internal forces and are not taken into consideration in the determination of the supporting forces for the structure as a whole. The distortion of any practically rigid structure under its working loads is so small as to be negligible when determining supporting forces. When the forces acting at the different joints in a built-up structure cannot be determined by dividing the structure up into parts, the structure is said to be statically indeterminate internally. A structure may be statically indeterminate internally and still be statically determinate externally. Fundamental Problems in Graphical Statics

A force may be represented by a straight line in a determined position, and its magnitude by the length of the straight line. The direction in which it acts may be indicated by an arrow. Polygon of Forces The parallelogram of two forces intersecting each other (see Figs. 3.1.4 and 3.1.5) leads directly to the graphic composition by means of the triangle of forces. In Fig. 3.1.10, R is called the closing side, and represents the resultant of the forces F1 and F2 in magnitude and direction. Its position is given by the point of application O. By means of repeated use of the triangle of forces and by omitting the

Fig. 3.1.11

Fig. 3.1.12

If in a closed polygon one of the forces is reversed in direction, this force becomes the resultant of all the others. Determination of Stresses in Members of a Statically Determinate Plane Structure with Loads at Rest

It will be assumed that the loads are applied at the joints of the structure, i.e., at the points where the different members are connected, and that the connections are pins with no friction. The stresses in the members must then be along lines connecting the pins, unless any member is loaded at more than two points by pin connections. Equilibrium In order that the whole structure should be in equilibrium, it is necessary that the external forces (loads and supports) shall form a balanced system. Graphical and analytical methods are both of service. Supporting Forces When the supporting forces are to be determined, it is not necessary to pay any attention to the makeup of the structure under consideration so long as it is practically rigid; the loads may be taken as they occur, or the resultant of the loads may be used instead. When the stresses in the members of the structure are being determined, the loads must be distributed at the joints where they belong. Method of Joints When all the external forces have been determined, any joint at which there are not more than two unknown forces may be taken and these unknown forces determined by the methods of the stress polygon, resolution, or moments. In Fig. 3.1.13, let O be the joint of a structure and F be the only known force; but let O1 and O2 be two members of the structure joined at O. Then the lines of action of the unknown forces are known and their magnitude may be determined (1) by a stress polygon which, for equilibrium, must close; (2) by resolution into H and V components, using the condition of equilibrium H  0, V  0; or (3) by moments, using any convenient point on the line of action of O1 and O2 and the condition of equilibrium M  0. No more than two unknown forces can be determined. In this manner, proceeding from joint to joint, the stresses in all the members of the truss can usually be determined if the structure is statically determinate internally.

3-6

MECHANICS OF SOLIDS

Method of Sections The structure may be divided into parts by passing a section through it cutting some of its members; one part may then be treated as a rigid body and the external forces acting upon it determined. Some of these forces will be the stresses in the members themselves. For example, let xx (Fig. 3.1.14) be a section taken through a truss loaded at P1, P2, and P3, and supported on rollers at S. As the whole truss is in equilibrium, any part of it must be also, and consequently the part shown to the left of xx must be in equilibrium under the action of the forces acting externally to it. Three of these forces are the

Quadrant, AB (Fig. 3.1.18a) x0  y0  2r/  0.6366r. Semicircumference, AC (Fig. 3.1.18b) y0  2r/  0.6366r; x0  0.

Y

Fig. 3.1.16

X

c

stresses in the members aa, bb, and bc, and are the unknown forces to be determined. They can be determined by applying the condition of equilibrium of forces acting in the same plane but not at the same point. H  0, V  0, M  0. The three unknown forces can be determined only if they are not parallel or do not pass through the same point; if, however, the forces are parallel or meet in a point, two unknown forces only can be determined. Sections may be passed through a structure cutting members in any convenient manner, as a rule, however, cutting not more than three members, unless members are unloaded. For the determination of stresses in framed structures, see Sec. 12.2. CENTER OF GRAVITY

Consider a three-dimensional body of any size, shape, and weight, but preferably thin and flat. If it is suspended as in Fig. 3.1.15 by a cord from any point A, it will be in equilibrium under the action of the tension in the cord and the weight W. If the experiment is repeated by suspending the body from point B, it will again be in equilibrium. Use a plumb bob through point A and draw the “first line” on the body. If the lines of action of the weight were marked in each case, they would be concurrent at a point G known as the center of gravity or center of mass. Whenever the density of the body is uniform, it will be a constant factor and like geometric shapes of different densities will have the same center of gravity. The term centroid is used in this case since the location of the center of gravity is of geometric concern only. If densities are nonuniform, like geometric shapes will have the same centroid but different centers of gravity.

r Y

Fig. 3.1.17

Fig. 3.1.18a

Y

X

r y0

D

C

First line W

X

Y

Fig. 3.1.18b

Fig. 3.1.19

CENTROIDS OF PLANE AREAS Triangle Centroid lies at the intersection of the lines joining the vertices with the midpoints of the sides, and at a distance from any side equal to one-third of the corresponding altitude. Parallelogram Centroid lies at the point of intersection of the diagonals. Trapezoid (Fig. 3.1.20) Centroid lies on the line joining the middle points m and n of the parallel sides. The distances ha and hb are h a 5 hsa 1 2bd/3sa 1 bd

h b 5 hs2a 1 bd/3sa 1 bd

Draw BE  a and CF  b; EF will then intersect mn at centroid. Any Quadrilateral The centroid of any quadrilateral may be determined by the general rule for areas, or graphically by dividing it into two sets of triangles by means of the diagonals. Find the centroid of each of the four triangles and connect the centroids of the triangles belonging to the same set. The intersection of these lines will be centroid of area. Thus, in Fig. 3.1.21, O, O1, O2, and O3 are, respectively, the centroids of the triangles ABD, ABC, BDC, and ACD. The intersection of O1O3 with OO2 gives the centroid of the quadrilateral area. B O1

A

A e

B

r

B

t lin

X x0

C X0

A

B s Fir

A

y0

X

Combination of Arcs and Straight Line (Fig. 3.1.19) AD and BC are two quadrants of radius r. y0  {(AB)r  2[0.5r(r  0.6366r)]} [AB  2(0.5r)] (Symmetrical about YY ).

Fig. 3.1.14

A

A c

X

Y

Fig. 3.1.13

Y B

CG

O

G

O3

D

W

Fig. 3.1.15 Centroids of Technically Important Lines, Areas, and Solids

CENTROIDS OF LINES Straight Lines The centroid is at its middle point. Circular Arc AB (Fig. 3.1.16) x0  r sin c/rad c; y0  2r sin2 1⁄2 c/rad c. (rad c  angle c measured in radians.) Circular Arc AC (Fig. 3.1.17) x0  r sin c/rad c; y0  0.

C

O2

Fig. 3.1.20

Fig. 3.1.21

Segment of a Circle (Fig. 3.1.22) x0  2⁄3 r sin3 c/(rad c  cos c sin c). A segment may be considered to be a sector from which a triangle is subtracted, and the general rule applied. Sector of a Circle (Fig. 3.1.23) x0  2⁄3 r sin c/rad c; y0  4⁄3 r sin2 1⁄2 c/rad c. Semicircle x0  4⁄3 r/  0.4244r; y0  0. Quadrant (90 sector) x0  y0  4⁄3 r/  0.4244r.

MOMENT OF INERTIA Parabolic Half Segment (Fig. 3.1.24) Area ABO: x0  3⁄5 x1; y0  3⁄8 y1. Parabolic Spandrel (Fig. 3.1.24) Area AOC: xr0 5 3⁄10 x 1; yr0 5 3⁄4y1.

Fig. 3.1.22

Truncated Circular Cone If h is the height of the frustum and R and r the radii of the bases, the distance from the surface of the base whose radius is R to the centroid is h(R2  2Rr  3r2)/4(R2  Rr  r2).

Fig. 3.1.23

Fig. 3.1.27

Segment of a Sphere (Fig. 3.1.28)

4(3r  h).

Quadrant of an Ellipse (Fig. 3.1.25) Area OAB: x0  4⁄3 (a/); y0  ⁄3 (b/). The centroid of a figure such as that shown in Fig. 3.1.26 may be determined as follows: Divide the area OABC into a number of parts by lines drawn perpendicular to the axis XX, e.g., 11, 22, 33, etc. These parts will be approximately either triangles, rectangles, or trapezoids. The area of each division may be obtained by taking the product of its

4

Y A b O

Fig. 3.1.28

Volume ABC: x0  3(2r  h)2/

Hemisphere x0  3r/8. Hollow Hemisphere x0  3(R4  r4)/8(R3  r3), where R and r are,

Fig. 3.1.24

X

3-7

x0

y0 a

B

X

Y

respectively, the outer and inner radii. Sector of a Sphere (Fig. 3.1.28) Volume OABCO: xr0 5 3⁄8 s2r 2 hd. Ellipsoid, with Semiaxes a, b, and c For each octant, distance from center of gravity to each of the bounding planes  3⁄8  length of semiaxis perpendicular to the plane considered. The formulas given for the determination of the centroid of lines and areas can be used to determine the areas and volumes of surfaces and solids of revolution, respectively, by employing the theorems of Pappus, Sec. 2.1. Determination of Center of Gravity of a Body by Experiment The center of gravity may be determined by hanging the body up from different points and plumbing down; the point of intersection of the plumb lines will give the center of gravity. It may also be determined as shown in Fig. 3.1.29. The body is placed on knife-edges which rest on platform scales. The sum of the weights registered on the two scales (w1  w2) must equal the weight (w) of the body. Taking a moment axis at either end (say, O), w2 A/w  x0  distance from O to plane containing the center of gravity. (See also Fig. 3.1.15 and accompanying text.)

Fig. 3.1.26

Fig. 3.1.25

mean height and its base. The centroid of each area may be obtained as previously shown. The sum of the moments of all the areas about XX and YY, respectively, divided by the sum of the areas will give approximately the distances from the center of gravity of the whole area to the axes XX and YY. The greater the number of areas taken, the more nearly exact the result.

Fig. 3.1.29

CENTROIDS OF SOLIDS Prism or Cylinder with Parallel Bases The centroid lies in the center of the line connecting the centers of gravity of the bases. Oblique Frustum of a Right Circular Cylinder (Fig. 3.1.27) Let 1 2 3 4 be the plane of symmetry. The distance from the base to the centroid is 1⁄2 h  (r2 tan2 c)/8h, where c is the angle of inclination of the oblique section to the base. The distance of the centroid from the axis of the cylinder is r2 tan c/4h. Pyramid or Cone The centroid lies in the line connecting the centroid of the base with the vertex and at a distance of one-fourth of the altitude above the base. Truncated Pyramid If h is the height of the truncated pyramid and A and B the areas of its bases, the distance of its centroid from the surface of A is

hsA 1 2 2AB 1 3Bd/4sA 1 2AB 1 Bd

Graphical Determination of the Centroids of Plane Areas

See Fig.

3.1.40. MOMENT OF INERTIA

The moment of inertia of a solid body with respect to a given axis is the limit of the sum of the products of the masses of each of the elementary particles into which the body may be conceived to be divided and the square of their distance from the given axis. If dm  dw/g represents the mass of an elementary particle and y its distance from an axis, the moment of inertia I of the body about this axis will be I  y2 dm  y2 dw/g. The moment of inertia may be expressed in weight units (Iw  y2 dw), in which case the moment of inertia in weight units, Iw, is equal to the moment of inertia in mass units, I, multiplied by g.

3-8

MECHANICS OF SOLIDS

If I  k2m, the quantity k is called the radius of gyration or the radius of inertia.

If a body is considered to be composed of a number of parts, its moment of inertia about an axis is equal to the sum of the moments of inertia of the several parts about the same axis, or I  I1  I2  I3      In. The moment of inertia of an area with respect to a given axis that lies in the plane of the area is the limit of the sum of the products of the elementary areas into which the area may be conceived to be divided and the square of their distance (y) from the axis in question. I  y2 dA  k2A, where k  radius of gyration. The quantity y2 dA is more properly referred to as the second moment of area since it is not a measure of inertia in a true sense. Formulas for moments of inertia and radii of gyration of various areas follow later in this section. Relation between the Moments of Inertia of an Area and a Solid

The moment of inertia of a solid of elementary thickness about an axis is equal to the moment of inertia of the area of one face of the solid about the same axis multiplied by the mass per unit volume of the solid times the elementary thickness of the solid. Moments of Inertia about Parallel Axes The moment of inertia of an area or solid about any given axis is equal to the moment of inertia about a parallel axis through the center of gravity plus the square of the distance between the two axes times the area or mass. In Fig. 3.1.30a, the moment of inertia of the area ABCD about axis YY is equal to I0 (or the moment of inertia about Y0Y0 through the center of gravity of the area and parallel to YY) plus x 20 A, where A  area of ABCD. In Fig. 3.1.30b, the moment of inertia of the mass m about YY 5 I0 1 x 20 m. Y0Y0 passes through the centroid of the mass and is parallel to YY.

Principal Moments of Inertia In every plane area, a given point being taken as the origin, there is at least one pair of rectangular axes

Fig. 3.1.31

Fig. 3.1.32

in the plane of the area about one of which the moment of inertia is a maximum, and a minimum about the other. These moments of inertia are called the principal moments of inertia, and the axes about which they are taken are the principal axes of inertia. One of the conditions for principal moments of inertia is that the product of inertia Ixy shall equal zero. Axes of symmetry of an area are always principal axes of inertia. Relation between Products of Inertia and Parallel Axes In Fig. 3.1.33, X0X0 and Y0Y0 pass through the center of gravity of the area parallel to the given axes XX and YY. If Ixy is the product of inertia for XX and YY, and Ix0 y0 that for X0X0 and Y0Y0, then Ixy 5 Ix0y0 1 abA. C.G.

Area A

Fig. 3.1.33 Mohr’s Circle The principal moments of inertia and the location of the principal axes of inertia for any point of a plane area may be established graphically as follows. Given at any point A of a plane area (Fig. 3.1.34), the moments of inertia Ix and Iy about axes X and Y, and the product of inertia Ixy relative to X and Y. The graph shown in Fig. 3.1.34b is plotted on rectangular coordinates with moments of inertia as abscissas and products of inertia

Fig. 3.1.30

Polar Moment of Inertia The polar moment of inertia (Fig. 3.1.31) is taken about an axis perpendicular to the plane of the area. Referring to Fig. 3.1.31, if Iy and Ix are the moments of inertia of the area A about YY and XX, respectively, then the polar moment of inertia Ip  Ix  Iy, or the polar moment of inertia is equal to the sum of the moments of inertia about any two axes at right angles to each other in the plane of the area and intersecting at the pole. Product of Inertia This quantity will be represented by Ixy, and is xy dy dx, where x and y are the coordinates of any elementary part into which the area may be conceived to be divided. Ixy may be positive or negative, depending upon the position of the area with respect to the coordinate axes XX and XY. Relation between Moments of Inertia about Axes Inclined to Each Other Referring to Fig. 3.1.32, let Iy and Ix be the moments of inertia

of the area A about YY and XX, respectively, Iry and Irx the moments about Y9Y9 and X9X9, and Ixy and Irxy the products of inertia for XX and YY, and X9X9 and Y9Y9, respectively. Also, let c be the angle between the respective pairs of axes, as shown. Then, Iry 5 Iy cos2 c 1 Ix sin2 c 1 Ixy sin 2c Irx 5 Ix cos2 c 1 Iy sin2 c 1 Ixy sin 2c Ix 2 Iy sin 2c 1 Ixy cos 2c Irxy 5 2

Fig. 3.1.34

as ordinates. Lay out Oa  Ix and ab  Ixy (upward for positive products of inertia, downward for negative). Lay out Oc  Iy and cd  negative of Ixy. Draw a circle with bd as diameter. This is Mohr’s circle. The maximum moment of inertia Irx 5 Of; the minimum moment of inertia is Iry 5 Og. The principal axes of inertia are located as follows. From axis AX (Fig. 3.1.34a) lay out angular distance   1⁄2 bef. This locates axis AX, one principal axis sIrx 5 Of d. The other principal axis of inertia is AY, perpendicular to AX sIrx 5 Ogd. The moment of inertia of any area may be considered to be made up of the sum or difference of the known moments of inertia of simple figures. For example, the dimensioned figure shown in Fig. 3.1.35 represents the section of a rolled shape with hole oprs and may be divided

MOMENT OF INERTIA

into the semicircle abc, rectangle edkg, and triangles mfg and hkl, from which the rectangle oprs is to be subtracted. Referring to axis XX, Ixx 5 p44/8 for semicircle abc 1 s2 3 113d/3 for rectangle edkg 1 2[s5 3 33d/36 1 102 s5 3 3d/2] for the two triangles mfg and hkl From the sum of these there is to be subtracted Ixx  [(2  32)/12  42(2  3)] for the rectangle oprs. If the moment of inertia of the whole area is required about an axis parallel to XX, but passing through the center of gravity of the whole area, I0 5 Ixx 2 x 20 A, where x0  distance from XX to center of gravity. The moments of inertia of built-up sections used in structural work may be found in the same manner, the moments of inertia of the Fig. 3.1.35 different rolled sections being given in Sec. 12.2. Moments of Inertia of Solids For moments of inertia of solids about parallel axes, Ix 5 I0 1 x 20 m. Moment of Inertia with Reference to Any Axis Let a mass particle dm of a body have x, y, and z as coordinates, XX, YY, and ZZ being the coordinate axes and O the origin. Let X9X9 be any axis passing through the origin and making angles of A, B, and C with XX, YY, and ZZ, respectively. The moment of inertia with respect to this axis then becomes equal to Irx 5 cos2 Asy 2 1 z 2d dm 1 cos 2 Bsz 2 3 x 2d dm 1 cos 2 Csx 2 1 y 2d dm 2 2 cos B cos Cyz dm 2 2 cos C cos Azx dm 2 2 cos A cos Bxy dm Let the moment of inertia about XX  Ix  (y2  z2) dm, about YY  Iy  (z2  x2) dm, and about ZZ  Iz  (x2  y2) dm. Let the products of inertia about the three coordinate axes be Iyz 5 yz dm

Izx 5 zx dm

3-9

Solid right circular cone about an axis through its apex and perpendicular to its axis: I  3M[(r2/4)  h2]/5. (h  altitude of cone, r  radius of base.) Solid right circular cone about its axis of revolution: I  3Mr2/10. Ellipsoid with semiaxes a, b, and c: I about diameter 2c (z axis)  4mabc (a2  b2)/15. [Equation of ellipsoid: (x2/a2)  (y2/b2)  (z2/c2)  1.] Ring with Circular Section (Fig. 3.1.36) Iyy  1⁄2m2Ra2(4R2  3a2); Ixx  m2Ra2[R2  (5a2/4)]. Y

Y

Fig. 3.1.36

x

dx 艎

Fig. 3.1.37

Approximate Moments of Inertia of Solids In order to determine the moment of inertia of a solid, it is necessary to know all its dimensions. In the case of a rod of mass M (Fig. 3.1.37) and length l, with shape and size of the cross section unknown, making the approximation that the weight is all concentrated along the axis of the rod, the moment l

of inertia about YY will be Iyy 5 3 sM/ldx 2 dx 5 Ml 2/3 0

A thin plate may be treated in the same way (Fig. 3.1.38): Iyy  l 1 2 2 3 sM/ldx dx 5 3 Ml . 0 Thin Ring, or Cylinder (Fig. 3.1.39) Assume the mass M of the ring or cylinder to be concentrated at a distance r from O. The moment of inertia about an axis through O perpendicular to plane of ring or along the axis of the cylinder will be I  Mr2; this will be greater than the exact moment of inertia, and r is sometimes taken as the distance from O to the center of gravity of the cross section of the rim.

Ixy 5 xy dm

Then the moment of inertia Irx becomes equal to Ix cos 2 A 1 Iy cos 2 B 1 Iz cos 2 C 2 2Iyz cos B cos C 2 2Izx cos C cos A 2 2Ixy cos A cos B The moment of inertia of any solid may be considered to be made up of the sum or difference of the moments of inertia of simple solids of which the moments of inertia are known. Moments of Inertia of Important Solids (Homogeneous)

w  weight per unit of volume of the body m  w/g  mass per unit of volume of the body M  W/g  total mass of body r  radius I  moment of inertia (mass units) Iw  I  g  moment of inertia (weight units) Solid circular cylinder about its axis: I  r4ma/2  Mr2/2. (a  length of axis of cylinder.) Solid circular cylinder about an axis through the center of gravity and perpendicular to axis of cylinder: I  M[r2  (a2/3)]/4. Hollow circular cylinder about its axis: I 5 Msr 21 1 r 22d/2 (r1 and r2  outer and inner radii). Thin-walled hollow circular cylinder about its axis: I  Mr2. Solid sphere about a diameter: I  8mr5/15  2Mr2/5. Thin hollow sphere about a diameter: I  2Mr2/3. Thick hollow sphere about a diameter: I 5 8mpsr 51 2 r 52d/15. (r1 and r2 are outer and inner radii.) Rectangular prism about an axis through center of gravity and perpendicular to a face whose dimensions are a and b: I  M(a2  b2)/12.

Fig. 3.1.38

Fig. 3.1.39

Flywheel Effect The moment of inertia of a solid is often called flywheel effect in the solution of problems dealing with rotating bodies. Graphical Determination of the Centroids and Moments of Inertia of Plane Areas Required to find the center of gravity of the area MNP

(Fig. 3.1.40) and its moment of inertia about any axis XX. Draw any line SS parallel to XX and at a distance d from it. Draw a number of lines such as AB and EF across the figure parallel to XX. From E and F draw ER and FT perpendicular to SS. Select as a pole any

Fig. 3.1.40

3-10

MECHANICS OF SOLIDS

point on XX, preferably the point nearest the area, and draw OR and OT, cutting EF at E9 and F9. If the same construction is repeated, using other lines parallel to XX, a number of points will be obtained, which, if connected by a smooth curve, will give the area MNP. Project E and F onto SS by lines ER and FT. Join F and T with O, obtaining E and F; connect the points obtained using other lines parallel to XX and obtain an area MNP. The area MNP  d  moment of area MNP about the line XX, and the distance from XX to the centroid MNP  area MNP  d/area MNP. Also, area MNP  d 2  moment of inertia of MNP about XX. The areas MNP9 and MNP can best be obtained by use of a planimeter. KINEMATICS Kinematics is the study of the motion of bodies without reference to the

forces causing that motion or the mass of the bodies. The displacement of a point is the directed distance that a point has moved on a geometric path from a convenient origin. It is a vector, having both magnitude and direction, and is subject to all the laws and characteristics attributed to vectors. In Fig. 3.1.41, the displacement of the point A from the origin O is the directed distance O to A, symbolized by the vector s. The velocity of a point is the time rate of change of displacement, or v  ds/dt. The acceleration of a point is the time rate of change of velocity, or a  dv/dt.

A velocity-time curve offers a convenient means for the study of acceleration. The slope of the curve at any point will represent the acceleration at that time. In Fig. 3.1.43a the slope is constant; so the acceleration must be constant. In the case represented by the full line, the acceleration is positive; so the velocity is increasing. The dotted line shows a negative acceleration and therefore a decreasing velocity. In Fig. 3.1.43b the slope of the curve varies from point to point; so the acceleration must also vary. At p and q the slope is zero; therefore, the acceleration of the point at the corresponding times must also be zero. The area under the velocity-time curve between any two ordinates such as NL and HT will represent the distance moved in time interval LT. In the case of the uniformly accelerated motion shown by the full line in Fig. 3.1.43a, the area LNHT is 1⁄2(NL  HT )  (OT  OL)  mean velocity multiplied by the time interval  space passed over during this time interval. In Fig. 3.1.43b the mean velocity can be obtained from the equation of the curve by means of the calculus, or graphically by approximation of the area.

Fig. 3.1.43

An acceleration-time curve (Fig. 3.1.44) may be constructed by plotting accelerations as ordinates, and times as abscissas. The area under this curve between any two ordinates will represent the total increase in velocity during the time interval. The area ABCD represents the total increase in velocity between time t1 and time t2. General Expressions Showing the Relations between Space, Time, Velocity, and Acceleration for Rectilinear Motion

SPECIAL MOTIONS

Fig. 3.1.41

The kinematic definitions of velocity and acceleration involve the four variables, displacement, velocity, acceleration, and time. If we eliminate the variable of time, a third equation of motion is obtained, ds/v  dt  dv/a. This differential equation, together with the definitions of velocity and acceleration, make up the three kinematic equations of motion, v  ds/dt, a  dv/dt, and a ds  v dv. These differential equations are usually limited to the scalar form when expressed together, since the last can only be properly expressed in terms of the scalar dt. The first two, since they are definitions for velocity and acceleration, are vector equations. A space-time curve offers a convenient means for the study of the motion of a point that moves in a straight line. The slope of the curve at any point will represent the velocity at that time. In Fig. 3.1.42a the slope is constant, as the graph is a straight line; the velocity is therefore uniform. In Fig. 3.1.42b the slope of the curve varies from point to point, and the velocity must also vary. At p and q the slope is zero; therefore, the velocity of the point at the corresponding times must also be zero.

Fig. 3.1.42

Uniform Motion If the velocity is constant, the acceleration must be zero, and the point has uniform motion. The space-time curve becomes a straight line inclined toward the time axis (Fig. 3.1.42a). The velocitytime curve becomes a straight line parallel to the time axis. For this motion a  0, v  constant, and s  s0  vt. Uniformly Accelerated or Retarded Motion If the velocity is not uniform but the acceleration is constant, the point has uniformly accelerated motion; the acceleration may be either positive or negative. The space-time curve becomes a parabola and the velocity-time curve becomes a straight line inclined toward the time axis (Fig. 3.1.43a). The acceleration-time curve becomes a straight line parallel to the time axis. For this motion a  constant, v  v0  at, s  s0  v0t  1⁄2 at2. If the point starts from rest, v0  0. Care should be taken concerning the sign  or  for acceleration. Composition and Resolution of Velocities and Acceleration Resultant Velocity A velocity is said to be the resultant of two other velocities when it is represented by a vector that is the geometric sum of the vectors representing the other two velocities. This is the parallelogram of motion. In Fig. 3.1.45, v is the resultant of v1 and v2

Fig. 3.1.44

Fig. 3.1.45

KINEMATICS

and is represented by the diagonal of a parallelogram of which v1 and v2 are the sides; or it is the third side of a triangle of which v1 and v2 are the other two sides. Polygon of Motion The parallelogram of motion may be extended to the polygon of motion. Let v1, v2, v3, v4 (Fig. 3.1.46a) show the directions of four velocities imparted in the same plane to point O. If the lines v1, v2, v3, v4 (Fig. 3.1.46b) are drawn parallel to and proportional to the velocities imparted to point O, v will represent the resultant velocity imparted to O. It will make no difference in what order the velocities are taken in constructing the motion polygon. As long as the arrows showing the direction of the motion follow each other in order about the polygon, the resultant velocity of the point will be represented in magnitude by the closing side of the polygon, but opposite in direction.

3-11

in the path is resolved by means of a parallelogram into components tangent and normal to the path, the normal acceleration an  v2/, where   radius of curvature of the path at the point in question, and the tangential acceleration at  dv/dt, where v  velocity tangent to the path at the same point. a 5 2a 2n 1 a 2t . The normal acceleration is constantly directed toward the concave side of the path.

Fig. 3.1.48

Fig. 3.1.46 Resolution of Velocities Velocities may be resolved into component velocities in the same plane, as shown by Fig. 3.1.47. Let the velocity of

EXAMPLE. Figure 3.1.49 shows a point moving in a curvilinear path. At p1 the velocity is v1; at p2 the velocity is v2. If these velocities are drawn from pole O (Fig. 3.1.49b), v will be the difference between v2 and v1. The acceleration during travel p1p2 will be v/t, where t is the time interval. The approximation becomes closer to instantaneous acceleration as shorter intervals t are employed.

point O be vr. In Fig. 3.1.47a this velocity is resolved into two components in the same plane as vr and at right angles to each other. vr 5 2sv1d2 1 sv2 d2 In Fig. 3.1.47b the components are in the same plane as vr , but are not at right angles to each other. In this case, vr 5 2sv1d2 1 sv2d2 1 2v1v2 cos B If the components v1 and v2 and angle B are known, the direction of vr can be determined. sin bOc  (v1/vr) sin B. sin cOa  (v2/vr) sin B. Where v1 and v2 are at right angles to each other, sin B  1.

Fig. 3.1.49

Fig. 3.1.47

Accelerations may be combined and resolved in the same manner as velocities, but in this case the lines or vectors represent accelerations instead of velocities. If the acceleration had components of magnitude a1 and a2, the magnitude of the resultant acceleration would be a 5 2sa1d2 1 sa2d2 1 2a1a2 cos B, where B is the angle between the vectors a1 and a2. Resultant Acceleration

Curvilinear Motion in a Plane

The linear velocity v  ds/dt of a point in curvilinear motion is the same as for rectilinear motion. Its direction is tangent to the path of the point. In Fig. 3.1.48a, let P1P2P3 be the path of a moving point and V1, V2, V3 represent its velocity at points P1, P2, P3, respectively. If O is taken as a pole (Fig. 3.1.48b) and vectors V1, V2, V3 representing the velocities of the point at P1, P2, and P3 are drawn, the curve connecting the terminal points of these vectors is known as the hodograph of the motion. This velocity diagram is applicable only to motions all in the same plane. Acceleration Tangents to the curve (Fig. 3.1.48b) indicate the directions of the instantaneous velocities. The direction of the tangents does not, as a rule, coincide with the direction of the accelerations as represented by tangents to the path. If the acceleration a at some point

The acceleration v/t can be resolved into normal and tangential components leading to an  vn/t, normal to the path, and ar  vp/t, tangential to the path.

Velocity and acceleration may be expressed in polar coordinates such that v 5 2v 2r 1 v 2u and a 5 2a 2r 1 a 2u. Figure 3.1.50 may be used to explain the r and  coordinates. EXAMPLE. At P1 the velocity is v1, with components v1r in the r direction and v1 in the  direction. At P2 the velocity is v2, with components v2r in the r direction and v2 in the  direction. It is evident that the difference in velocities v2  v1  v will have components vr and v, giving rise to accelerations ar and a in a time interval t.

In polar coordinates, vr  dr/dt, ar  d 2r/dt2  r(d/dt)2, v  r(d/dt), and a  r(d 2/dt2)  2(dr/dt)(d/dt). If a point P moves on a circular path of radius r with an angular velocity of  and an angular acceleration of , the linear velocity of the point P is v  r and the two components of the linear acceleration are an  v2/r  2r  v and at  r. If the angular velocity is constant, the point P travels equal circular paths in equal intervals of time. The projected displacement, velocity, and acceleration of the point P on the x and y axes are sinusoidal functions of time, and the motion is said to be harmonic motion. Angular velocity is usually expressed in radians per second, and when the number (N) of revolutions traversed per minute (N/min) by the point P is known, the angular velocity of the radius r is   2N/60  0.10472N.

3-12

MECHANICS OF SOLIDS

Fig. 3.1.50

Fig. 3.1.51

In Fig. 3.1.51, let the angular velocity of the line OP be a constant . Let the point P start at X9 and move to P in time t. Then the angle   t. If OP  r, OA 5 s 5 r cos vt. The velocity v of the point A on the x axis will equal ds/dt  r sin t, and the acceleration a  dv/dt  2r cos t. The period  is the time necessary for the point P to complete one cycle of motion   2/, and it is also equal to the time necessary for A to complete a full cycle on the x axis from X to X and return.

line and relating the motion of all other parts of the rigid body to these motions. If a rigid body moves so that a straight line connecting any two of its particles remains parallel to its original position at all times, it is said to have translation. In rectilinear translation, all points move in straight lines. In curvilinear translation, all points move on congruent curves but without rotation. Rotation is defined as angular motion about an axis, which may or may not be fixed. Rigid body motion in which the paths of all particles lie on parallel planes is called plane motion.

Curvilinear Motion in Space

Angular Motion

If three dimensions are used, velocities and accelerations may be resolved into components not in the same plane by what is known as the parallelepiped of motion. Three coordinate systems are widely used, cartesian, cylindrical, and spherical. In cartesian coordinates, v 5 2v 2x 1 v 2y 1 v 2z and a 5 2a 2x 1 a 2y 1 a 2z . In cylindrical coordinates, the radius vector R of displacement lies in the rz plane, which is at an angle with the xz plane. Referring to (a) of Fig. 3.1.52, the  coordinate is perpendicular to the rz plane. In this system v 5 2v 2r 1 v 2u 1 v 2z and a 5 2a 2r 1 a 2u 1 a 2z where vr  dr/dt, ar  d 2r/dt2  r(d/dt)2, v  r(d/dt), and a  r(d 2/dt2)  2(dr/dt)(d/dt). In spherical coordinates, the three coordinates are the R coordinate, the  coordinate, and the  coordinate as in (b) of Fig. 3.1.52. The velocity and acceleration are v 5 2v 2R 1 v 2u 1 v 2f

Angular displacement is the change in angular position of a given line as

measured from a convenient reference line. In Fig. 3.1.53, consider the motion of the line AB as it moves from its original position A9B9. The angle between lines AB and A9B9 is the angular displacement of line AB, symbolized as . It is a directed quantity and is a vector. The usual notation used to designate angular displacement is a vector normal to

and a 5 2a 2R 1 a 2u 1 a 2f, where vR  dR/dt, vf  R(df/dt), vu  R cos f(du/dt), aR  d 2R/dt 2  R(df/dt)2  R cos2 f(du/dt)2, af  R(d 2f/dt 2)  R cos f sin f (du/dt)2  2(dR/dt)(df/dt), and au  R cos f (d 2u/dt 2)  2[(dR/dt) cos f  R sin f (df/dt)] du/dt. Fig. 3.1.53

Fig. 3.1.52 Motion of Rigid Bodies

A body is said to be rigid when the distances between all its particles are invariable. Theoretically, rigid bodies do not exist, but materials used in engineering are rigid under most practical working conditions. The motion of a rigid body can be completely described by knowing the angular motion of a line on the rigid body and the linear motion of a point on this

the plane in which the angular displacement occurs. The length of the vector is proportional to the magnitude of the angular displacement. For a rigid body moving in three dimensions, the line AB may have angular motion about any three orthogonal axes. For example, the angular displacement can be described in cartesian coordinates as   x  y  z , where u 5 2u 2x 1 u 2y 1 u 2z . Angular velocity is defined as the time rate of change of angular displacement,   d/dt. Angular velocity may also have components about any three orthogonal axes. Angular acceleration is defined as the time rate of change of angular velocity,   d/dt  d 2 dt2. Angular acceleration may also have components about any three orthogonal axes. The kinematic equations of angular motion of a line are analogous to those for the motion of a point. In referring to Table 3.1.1,   d/dt   d/dt, and  d   d. Substitute  for s,  for v, and  for a. Motion of a Rigid Body in a Plane Plane motion is the motion of a rigid body such that the paths of all par-

ticles of that rigid body lie on parallel planes.

KINEMATICS

3-13

Table 3.1.1 s  f(t)

Variables

v  f(t) t

s 5 s0 1 3 v dt

Displacement

a2  f(t) t

s 5 s0 1 3 3 a dt dt

t0

Velocity

v  ds/dt

Acceleration

a  d 2 s/dt2

t

t0 t0 t

v 5 v0 1 3 a dt t0

a  dv/dt

a  f(s, v) v

s 5 s0 1 3 sv/ad dv v

v0 s0

v0

s

3 v dv 5 3 a ds a  v dv/ds

Instantaneous Axis When the axis about which any body may be considered to rotate changes its position, any one position is known as an instantaneous axis, and the line through all positions of the instantaneous axis as the centrode. When the velocity of two points in the same plane of a rigid body having plane motion is known, the instantaneous axis for the body will be at the intersection of the lines drawn from each point and perpendicular to its velocity. See Fig. 3.1.54, in which A and B are two points on the rod AB, v1 and v2 representing their velocities. O is the instantaneous axis for AB; therefore point C will have velocity shown in a line perpendicular to OC. Linear velocities of points in a body rotating about an instantaneous axis are proportional to their distances from this axis. In Fig. 3.1.54, v1: v2: v3  AO : OB : OC. If the velocities of A and B were parallel, the lines OA and OB would also be parallel and there would be no instantaneous axis. The motion of the rod would be translation, and all points would be moving with the same velocity in parallel straight lines. If a body has plane motion, the components of the velocities of any two points in the body along the straight line joining them must be equal. Ax

must be equal to By and Cz in Fig. 3.1.54. EXAMPLE. In Fig. 3.1.55a, the velocities of points A and B are known—they are v1 and v2, respectively. To find the instantaneous axis of the body, perpendiculars AO and BO are drawn. O, at the intersection of the perpendiculars, is the instantaneous axis of the body. To find the velocity of any other point, like C, line OC is drawn and v3 erected perpendicular to OC with magnitude equal to v1 (CO/AO). The angular velocity of the body will be   v1/AO or v2/BO or v3/CO. The instantaneous axis of a wheel rolling on a rack without slipping (Fig. 3.1.55b) lies at the point of contact O, which has zero linear velocity. All points of the wheel will have velocities perpendicular to radii to O and proportional in magnitudes to their respective distances from O.

Another way to describe the plane motion of a rigid body is with the use of relative motion. In Fig. 3.1.56 the velocity of point A is v1. The angular velocity of the line AB is AB. The velocity of B relative to A is AB  rAB. Point B is considered to be moving on a circular path around A as a center. The direction of relative velocity of B to A would be tangent to the circular path in the direction that AB would make B move. The velocity of B is the vector sum of the velocity A added to the velocity of B relative to A, vB  vA  vB/A. The acceleration of B is the vector sum of the acceleration of A added to the acceleration of B relative to A, aB  aA  aB/A. Care must be taken to include the complete relative acceleration of B to A. If B is considered to move on a circular path about A, with a velocity relative to A, it will have an acceleration relative to A that has both normal and tangential components: aB/A  (aB/A)n  (aB/A)t.

Fig. 3.1.56

If B is a point on a path which lies on the same rigid body as the line AB, a particle P traveling on the path will have a velocity vp at the instant P passes over point B such that vP  vA  vB/A  vP/B, where the velocity vP/B is the velocity of P relative to point B. The particle P will have an acceleration aP at the instant P passes over the point B such that aP  aA  aB/A  aP/B  2AB  vP/B. The term aP/B is the acceleration of P relative to the path at point B. The last term 2AB vP/B is frequently referred to as the coriolis acceleration. The direction is always normal to the path in a sense which would rotate the head of the vector vP/B about its tail in the direction of the angular velocity of the rigid body AB. EXAMPLE. In Fig. 3.1.57, arm AB is rotating counterclockwise about A with a constant angular velocity of 38 r/min or 4 rad/s, and the slider moves outward with a velocity of 10 ft/s (3.05 m/s). At an instant when the slider P is 30 in (0.76 m) from the center A, the acceleration of the slider will have two components. One component is the normal acceleration directed toward the center A. Its magnitude is 2r  42 (30/12)  40 ft/s2 [2r  42 (0.76)  12.2 m/s2]. The second is the coriolis acceleration directed normal to the arm AB, upward and to the left. Its magnitude is 2v  2(4)(10)  80 ft/s2 [2v  2(4)(3.05)  24.4 m/s2].

Fig. 3.1.57 General Motion of a Rigid Body Fig. 3.1.54

Fig. 3.1.55

The general motion of a point moving in a coordinate system which is itself in motion is complicated and can best be summarized by using

3-14

MECHANICS OF SOLIDS

vector notation. Referring to Fig. 3.1.58, let the point P be displaced a vector distance R from the origin O of a moving reference frame x, y, z which has a velocity vo and an acceleration ao. If point P has a velocity and an acceleration relative to the moving reference plane, let these be vr and ar. The angular velocity of the moving reference fame is , and

Fig. 3.1.58

plane is (3/5)(90)  (4/5)(36)  9.36  15.84 lbf (70.46 N) downward. F  (W/9) a  (90/g) a; therefore, a  0.176 g  5.66 ft/s2 (1.725 m/s2). In SI units, F  ma  70.46  40.8a; and a  1.725 m/s2. The body is acted upon by 5

constant forces and starts from rest; therefore, v 5 3 a dt, and at the end of 5 s, 0 the velocity would be 28.35 ft/s (8.91 m/s). EXAMPLE 2. The force with which a rope acts on a body is equal and opposite to the force with which the body acts on the rope, and each is equal to the tension in the rope. In Fig. 3.1.60a, neglecting the weight of the pulley and the rope, the tension in the cord must be the force of 27 lbf. For the 18-lb mass, the unbalanced force is 27  18  9 lbf in the upward direction, i.e., 27  18  (18/g)a, and a  16.1 ft/s2 upward. In Fig. 3.1.60b the 27-lb force is replaced by a 27-lb mass. The unbalanced force is still 27  18  9 lbf, but it now acts on two masses so that 27  18  (45a/g) and a  6.44 ft/s2. The 18-lb mass is accelerated upward, and the 27-lb mass is accelerated downward. The tension in the rope is equal to 18 lbf plus the unbalanced force necessary to give it an upward acceleration of g/5 or T  18  (18/g)(g/5)  21.6 lbf. The tension is also equal to 27 lbf less the unbalanced force necessary to give it a downward acceleration of g/5 or T  27  (27/g)  (g/5)  21.6 lbf.

the origin of the moving reference frame is displaced a vector distance R1 from the origin of a primary (fixed) reference frame X, Y, Z. The velocity and acceleration of P are vP  vo    R  vr and aP  ao  (d/dt)  R    (  R)  2  vr  ar. DYNAMICS OF PARTICLES

Consider a particle of mass m subjected to the action of forces F1, F2, F3, . . . , whose vector resultant is R  F. According to Newton’s first law of motion, if R  0, the body is acted on by a balanced force system, and it will either remain at rest or move uniformly in a straight line. If R 2 0, Newton’s second law of motion states that the body will accelerate in the direction of and proportional to the magnitude of the resultant R. This may be expressed as F  ma. If the resultant of the force system has components in the x, y, and z directions, the resultant acceleration will have proportional components in the x, y, and z direction so that Fx  max, Fy  may, and Fz  maz. If the resultant of the force system varies with time, the acceleration will also vary with time. In rectilinear motion, the acceleration and the direction of the unbalanced force must be in the direction of motion. Forces must be in balance and the acceleration equal to zero in any direction other than the direction of motion. EXAMPLE 1. The body in Fig. 3.1.59 has a mass of 90 lbm (40.8 kg) and is subjected to an external horizontal force of 36 lbf (160 N) applied in the direction shown. The coefficient of friction between the body and the inclined plane is 0.1. Required, the velocity of the body at the end of 5 s, if it starts from rest.

(a)

(b)

Fig. 3.1.60 In SI units, in Fig. 3.1.60a, the unbalanced force is 120  80  40 N, in the upward direction, i.e., 120  80  8.16a, and a  4.9 m/s2 (16.1 ft/s2). In Fig. 3.1.60b the unbalanced force is still 40 N, but it now acts on the two masses so that 120  80  20.4a and a  1.96 m/s2 (6.44 ft/s2). The tension in the rope is the weight of the 8.16-kg mass in newtons plus the unbalanced force necessary to give it an upward acceleration of 1.96 m/s2, T  9.807(8.16)  (8.16)(1.96)  96 N (21.6 lbf ). General Formulas for the Motion of a Body under the Action of a Constant Unbalanced Force

Let s  distance, ft; a  acceleration, ft/s2; v  velocity, ft/s; v0  initial velocity, ft/s; h  height, ft; F  force; m  mass; w  weight; g  acceleration due to gravity. Initial velocity  0 F  ma  (w/g)a v  at s  1⁄2 at2  1⁄2 vt v 5 22as 5 22gh sfalling freely from restd

Fig. 3.1.59 First determine all the forces acting externally on the body. These are the applied force F  36 lbf (106 N), the weight W  90 lbf (400 N), and the force with which the plane reacts on the body. The latter force can be resolved into component forces, one normal and one parallel to the surface of the plane. Motion will be downward along the plane since a static analysis will show that the body will slide downward unless the static coefficient of friction is greater than 0.269. In the direction normal to the surface of the plane, the forces must be balanced. The normal force is (3/5)(36)  (4/5)(90)  93.6 lbf (416 N). The frictional force is 93.6  0.1  9.36 lbf (41.6 N). The unbalanced force acting on the body along the

Initial velocity  v0 F  ma  (w/g)a v  v0  at s  v0t  1⁄2 at2  1⁄2 v0t  1⁄2 vt If a body is to be moved in a straight line by a force, the line of action of this force must pass through its center of gravity. General Rule for the Solution of Problems When the Forces Are Constant in Magnitude and Direction

Resolve all the forces acting on the body into two components, one in the direction of the body’s motion and one at right angles to it. Add the

DYNAMICS OF PARTICLES

components in the direction of the body’s motion algebraically and find the unbalanced force, if any exists. In curvilinear motion, a particle moves along a curved path, and the resultant of the unbalanced force system may have components in directions other than the direction of motion. The acceleration in any given direction is proportional to the component of the resultant in that direction. It is common to utilize orthogonal coordinate systems such as cartesian coordinates, polar coordinates, and normal and tangential coordinates in analyzing forces and accelerations. EXAMPLE. A conical pendulum consists of a weight suspended from a cord or light rod and made to rotate in a horizontal circle about a vertical axis with a constant angular velocity of N r/min. For any given constant speed of rotation, the angle , the radius r, and the height h will have fixed values. Looking at Fig. 3.1.61, we see that the forces in the vertical direction must be balanced, T cos   w. The forces in the direction normal to the circular path of rotation are unbalanced such that T sin   (w/g)an  (w/g)2r. Substituting r  l sin  in this last equation gives the value of the tension in the cord T  (w/g)l2. Dividing the second equation by the first and substituting tan   r/h yields the additional relation that h  g/2.

3-15

to the body to constantly deviate it toward the axis. This deviating force is known as centripetal force. The equal and opposite resistance offered by the body to the connection is called the centrifugal force. The acceleration toward the axis necessary to keep a particle moving in a circle about that axis is v2/r; therefore, the force necessary is ma  mv2/r  wv2/gr  w2N2r/900g, where N  r/min. This force is constantly directed toward the axis. The centrifugal force of a solid body revolving about an axis is the same as if the whole mass of the body were concentrated at its center of gravity.

Centrifugal force  wv2/gr  mv2/r  w2r/g, where w and m are the weight and mass of the whole body, r is the distance from the axis about which the body is rotating to the center of gravity of the body,  the angular velocity of the body about the axis in radians, and v the linear velocity of the center of gravity of the body. Balancing

A rotating body is said to be in standing balance when its center of gravity coincides with the axis upon which it revolves. Standing balance may be obtained by resting the axle carrying the body upon two horizontal plane surfaces, as in Fig. 3.1.63. If the center of gravity of the wheel A lies on the axis of the shaft B, there will be no movement, but if the center of gravity does not lie on the axis of the shaft, the shaft will roll until the center of gravity of the wheel comes directly under the

Fig. 3.1.63

Fig. 3.1.61

An unresisted projectile has a motion compounded of the vertical motion of a falling body, and of the horizontal motion due to the horizontal component of the velocity of projection. In Fig. 3.1.62 the only force acting after the projectile starts is gravity, which causes an accelerating downward. The horizontal component of the original velocity v0 is not changed by gravity. The projectile will rise until the velocity

axis of the shaft. The center of gravity may be brought to the axis of the shaft by adding or taking away weight at proper points on the diameter passing through the center of gravity and the center of the shaft. Weights may be added to or subtracted from any part of the wheel so long as its center of gravity is brought to the center of the shaft. A rotating body may be in standing balance and not in dynamic balance. In Fig. 3.1.64, AA and BB are two disks whose centers of gravity are at o and p, respectively. The shaft and the disks are in standing balance if the disks are of the same weight and the distances of o and p from the center of the shaft are equal, and o and p lie in the same axial plane but on opposite sides of the shaft. Let the weight of each disk be w and the distances of o and p from the center of the shaft each be equal to r.

Fig. 3.1.62

given to it by gravity is equal to the vertical component of the starting velocity v0, and the equation v0 sin   gt gives the time t required to reach the highest point in the curve. The same time will be taken in falling if the surface XX is level, and the projectile will therefore be in flight 2t s. The distance s  v0 cos   2t, and the maximum height of ascent h  (v0 sin u)2/2g. The expressions for the coordinates of any point on the path of the projectile are: x  (v0 cos )t, and y  (v0 sin )t  1⁄2 gt2, giving y 5 x tan u 2 sgx 2/2v 2 cos 2 ud as the equation for the 0 curve of the path. The radius of curvature of the highest point may be found by using the general expression v2  gr and solving for r, v being taken equal to v0 cos . Simple Pendulum The period of oscillation  t 5 2p 2l/g, where l is the length of the pendulum and the length of the swing is not great compared to l. Centrifugal and Centripetal Forces When a body revolves about an axis, some connection must exist capable of applying force enough

Fig. 3.1.64

The force exerted on the shaft by AA is equal to w2r/g, where  is the angular velocity of the shaft. Also, the force exerted on the shaft by BB  w2r/g. These two equal and opposite parallel forces act at a distance x apart and constitute a couple with a moment tending to rotate the shaft, as shown by the arrows, of (w2r/g)x. A couple cannot be balanced by a single force; so two forces at least must be added to or subtracted from the system to get dynamic balance. Systems of Particles The principles of motion for a single particle can be extended to cover a system of particles. In this case, the vector resultant of all external forces acting on the system of particles must equal the total mass of the system times the acceleration of the mass center, and the direction of the resultant must be the direction of the acceleration of the mass center. This is the principle of motion of the mass center.

3-16

MECHANICS OF SOLIDS

Rotation of Solid Bodies in a Plane about Fixed Axes

For a rigid body revolving in a plane about a fixed axis, the resultant moment about that axis must be equal to the product of the moment of inertia (about that axis) and the angular acceleration, M0  I0. This is a

general statement which includes the particular case of rotation about an axis that passes through the center of gravity. Rotation about an Axis Passing through the Center of Gravity

The rotation of a body about its center of gravity can only be caused or changed by a couple. See Fig. 3.1.65. If a single force F is applied to the wheel, the axis immediately acts on the wheel with an equal force to prevent translation, and the result is a couple (moment Fr) acting on the body and causing rotation about its center of gravity.

Center of Percussion The distance from the axis of suspension to the center of percussion is q0 5 I/mrG, where I  moment of inertia of the body about its axis of suspension and rG is the distance from the axis of suspension to the center of gravity of the body. EXAMPLES. 1. Find the center of percussion of the homogeneous rod (Fig. 3.1.67) of length L and mass m, suspended at XX. 1 q0 5 mr G I
E pdsDo 1 Did> sDo 2 Did]

p 5 2rcV 5 2s1.937ds4,860ds210d 5 94,138 lbf/ft 2 5 94,138/144 5 653.8 lbf/in2 s4.507 3 106 N/m2d

At t 2: Q 2 5 0.61 3 0.08727 22 3 32.17 3 8 5 1.208 ft 3/s t2 2 t1 5

3-61

1 2 1.350 b 1 1.350 2 1.208d 1 2 1.208

t 2 2 t 1 5 205.4 s

Es Å r[1 1 sE s >E pdsDo 1 Did>sDo 2 Did] 319,000 3 144

5 ã

1.937c1 1

s319,000/28.5 3 106ds3.500 1 3.067d s3.500 2 3.067d

d

5 4,504 p 5 2s1.937ds4,504ds210d 5 87,242 lbf/ft 2 5 605.9 lbf/in2 s4.177 3 106 N/m2d

WATER HAMMER Equations Water hammer is the series of shocks, sounding like hammer blows, produced by suddenly reducing the flow of a fluid in a pipe. Consider a fluid flowing frictionlessly in a rigid pipe of uniform area A with a velocity V. The pipe has a length L, and inlet pressure p1 and a pressure p2 at L. At length L, there is a valve which can suddenly reduce the velocity at L to V  V. The equivalent mass rate of flow of a pressure wave traveling at sonic velocity c, M 5 rAc. From the impulse-momentum equation, MsV2 2 V1d 5 p2 A2 2 p1A1; for this application, ( rAc)(V  V  V )  p2 A  p1 A, or the increase in pressure p  cV. When the liquid is flowing in an elastic pipe, the

3.4

3. Maximum time for closure t 5 2L /c 5 2 3 200/4,860 5 0.08230 s or less than 1/10 s COMPUTATIONAL FLUID DYNAMICS (CFD)

The partial differential equations of fluid motion are very difficult to solve. CFD methods are utilized to provide discrete approximations for the solution of those equations. A brief introduction to CFD is included in Section 20.6. The references therein will guide the reader further.

VIBRATION

by Leonard Meirovitch REFERENCES: Harris, “Shock and Vibration Handbook,” 3d ed., McGraw-Hill. Thomson, “Theory of Vibration with Applications,” Prentice Hall. Meirovitch, “Fundamentals of Vibration,” McGraw-Hill. Meirovitch, “Principles and Techniques of Vibrations,” Prentice-Hall. SINGLE-DEGREE-OF-FREEDOM SYSTEMS Discrete System Components A system is defined as an aggregation of components acting together as one entity. The components of a vibratory mechanical system are of three different types, and they relate forces to displacements, velocities, and accelerations. The component relating forces to displacements is known as a spring (Fig. 3.4.1a). For a linear spring the force Fs is proportional to the elongation d 5 x 2 2 x 1, or

Fs 5 kd 5 ksx 2 2 x 1d

(3.4.1)

where k represents the spring constant, or the spring stiffness, and x 1 and x 2 are the displacements of the end points. The component relating

forces to velocities is called a viscous damper or a dashpot (Fig. 3.4.1b). It consists of a piston fitting loosely in a cylinder filled with liquid so that the liquid can flow around the piston when it moves relative to the cylinder. The relation between the damper force and the velocity of the piston relative to the cylinder is # # Fd 5 csx 2 2 x 1d (3.4.2) in which c is the coefficient of viscous damping; note that dots denote derivatives with respect to time. Finally, the relation between forces and accelerations is given by Newton’s second law of motion: $ Fm 5 mx (3.4.3) where m is the mass (Fig. 3.4.1c). The spring constant k, coefficient of viscous damping c, and mass m represent physical properties of the components and are the system parameters. By implication, these properties are concentrated at points,

3-62

VIBRATION

thus they are lumped, or discrete, parameters. Note that springs and dampers are assumed to be massless and masses are assumed to be rigid. Springs can be arranged in parallel and in series. Then, the proportionality constant between the forces and the end points is known as an

Table 3.4.1

Equivalent Spring Constants

Fig. 3.4.1

equivalent spring constant and is denoted by keq, as shown in Table 3.4.1.

Certain elastic components, although distributed over a given line segment, can be regarded as lumped with an equivalent spring constant given by keq  F/ , where is the deflection at the point of application of the force F. A similar relation can be given for springs in torsion. Table 3.4.1 lists the equivalent spring constants for a variety of components. Equation of Motion The dynamic behavior of many engineering systems can be approximated with good accuracy by the mass-damperspring model shown in Fig. 3.4.2. Using Newton’s second law in conjunction with Eqs. (3.4.1)–(3.4.3) and measuring the displacement x(t) from the static equilibrium position, we obtain the differential equation of motion $ # mx std 1 cx std 1 kxstd 5 Fstd (3.4.4) # which is subject to the initial conditions x(0)  x0, x s0d 5 v0, where x0 and v0 are the initial displacement and initial velocity, respectively. Equation (3.4.4) is in terms of a single coordinate. namely x(t); the system of Fig. 3.4.2 is therefore said to be a single-degree-of-freedom system. Free Vibration of Undamped Systems Assuming zero damping and external forces and dividing Eq. (3.4.4) through by m, we obtain $ x 1 v2nx 5 0 vn 5 2k/m (3.4.5) In this case, the vibration is caused by the initial excitations alone. The solution of Eq. (3.4.5) is xstd 5 A cos svnt 2 fd

(3.4.6)

phase angle do depend on the initial displacement and velocity, as follows: A 5 2x 20 1 sv0 /vnd2

rad/s

T 5 2p/vn

seconds

(3.4.9)

The reciprocal of the period provides another definition of the natural frequency, namely, fn 5

vn 1 5 T 2p

Hz

(3.4.10)

where Hz denotes hertz [1 Hz  1 cycle per second (cps)]. A large variety of vibratory systems behave like harmonic oscillators, many of them when restricted to small amplitudes. Table 3.4.2 shows a variety of harmonic oscillators together with their respective natural frequency. Free Vibration of Damped Systems Let F(t)  0 and divide through by m. Then, Eq. (3.4.4) reduces to $ # x std 1 2zvn x std 1 v2n xstd 5 0 (3.4.11) where z 5 c/2mvn (3.4.12) is the damping factor, a nondimensional quantity. The nature of the motion depends on z. The most important case is that in which 0 z 1.

(3.4.7)

Systems described by equations of the type (3.4.5) are called harmonic oscillators. Because the frequency of oscillation represents an inher-

ent property of the system, independent of the initial excitation, n is called the natural frequency. On the other hand, the amplitude and

(3.4.8)

The time necessary to complete one cycle of motion defines the period

which represents simple sinusoidal, or simple harmonic oscillation with amplitude A, phase angle f, and frequency vn 5 2k/m

f 5 tan 21v0 /x 0vn

Fig. 3.4.2

SINGLE-DEGREE-OF-FREEDOM SYSTEMS Table 3.4.2

Harmonic Oscillators and Natural Frequencies

3-63

In this case, the system is said to be underdamped and the solution of Eq. (3.4.11) is where

xstd 5 Ae2zvnt cos svd t 2 fd vd 5 s1 2 z2d1/2vn

(3.4.13) (3.4.14)

is the frequency of damped free vibration and T 5 2p/vd

(3.4.15)

is the period of damped oscillation. The amplitude and phase angle depend on the initial displacement and velocity, as follows: A 5 2x 20 1 szvn x 0 1 v0d2/v2d f 5 tan21 szvn x 0 1 v0d/x 0vd

(3.4.16)

The motion described by Eq. (3.4.13) represents decaying oscillation, where the term Ae2zvnt can be regarded as a time-dependent amplitude, providing an envelope bounding the harmonic oscillation. When z  1, the solution represents aperiodic decay. The case z  1 represents critical damping, and cc 5 2mvn

(3.4.17)

is the critical damping coefficient, although there is nothing critical about it. It merely represents the borderline between oscillatory decay and aperiodic decay. In fact, cc is the smallest damping coefficient for which the motion is aperiodic. When z  1, the system is said to be overdamped. Logarithmic Decrement

Quite often the damping factor is not known and must be determined experimentally. In the case in which the system is underdamped, this can be done conveniently by plotting x(t) versus t (Fig. 3.4.3) and measuring the response at two different times

Fig. 3.4.3

separated by a complete period. Let the times be t1 and t1  T, introduce the notation x(t1)  x1, x(t1  T)  x2, and use Eq. (3.4.13) to obtain Ae2zvnt1 cos svdt 1 2 fd x1 zvnT x 2 5 Ae2zvnst11Td cos [v st 1 Td 2 f] 5 e d 1

(3.4.18)

where cos [vd (t1  T )  f]  cos (vd t1  f  2p)  cos (vd t1  f). Equation (3.4.18) yields the logarithmic decrement x1 d 5 In x 5 zvnT 5 2

2pz

21 2 z2 which can be used to obtain the damping factor d z5 2s2pd2 1 d2

(3.4.19)

(3.4.20)

For small damping, the logarithmic decrement is also small, and the damping factor can be approximated by z
h. Then, considering the two linear interpolation functions f1 sjd 5 j

f2 sjd 5 1 2 j

(3.4.150)

0 0

the displacement at point  can be expressed as vsjd 5 ae21f1 sjd 1 aef2 sjd

dfi dfj 1 1 EA dj 3 h 0 dj dj

1

m eij 5 h 3 mfi fj dj, 0

i, j 5 1, 2

0 0

0 0

c c

2 21

21 Kh/EA (3.4.154)

(3.4.151)

where ae1 and ae are the nodal displacements for element e. Using Eqs. (3.4.143) and changing variables from x to , we can write the element stiffness and mass coefficients k eij 5

2 21 0 c 0 0 21 2 21 c 0 0 0 21 2 c 0 0 EA K5 F V h ..............................

(3.4.152)

4 1 0 c 0 0 1 4 1 c 0 0 hm 0 1 4 c 0 0 M5 F V 6 .................... 0 0 0 c 4 1 0 0 0 c 1 2

3-78

VIBRATION

For beams in bending, the displacements consist of one translation and one rotation per node; the interpolation functions are the Hermite cubics f1 sjd 5 3j2 2 2j3, f2 sjd 5 j2 2 j3 f3 sjd 5 1 2 3j2 1 2j3, f4 sjd 5 2j 1 2j2 2 j3 (3.4.155) and the element stiffness and mass coefficients are 2 2 1 1 1 d fi d fj dj meij 5 h 3 mfifj dj k eij 5 3 3 EI 2 2 h dj dj 0

0

i, j 5 1, 2, 3, 4

(3.4.156)

yielding typical element stiffness and mass matrices 12 6 EI Ke 5 3 D h 212 6

6 4 26 2

212 26 12 26

6 2 T 26 4

156 22 hm Me 5 D 420 54 213

22 4 13 23

54 13 156 222

213 23 T 222 4

Fig. 3.4.29

(3.4.157)

The treatment of two-dimensional problems, such as for membranes and plates, is considerably more complex (see Meirovitch, “Principles and Techniques of Vibration,” Prentice-Hall) than for one-dimensional problems. The various steps involved in the finite element method lend themselves to ready computer programming. There are many computer codes available commercially; one widely used is NASTRAN. VIBRATION-MEASURING INSTRUMENTS

Typical quantities to be measured include acceleration, velocity, displacement, frequency, damping, and stress. Vibration implies motion, so that there is a great deal of interest in transducers capable of measuring motion relative to the inertial space. The basic transducer of many vibrationmeasuring instruments is a mass-damper-spring enclosed in a case together with a device, generally electrical, for measuring the displacement of the mass relative to the case, as shown in Fig. 3.4.29. The equation for the displacement z(t) of the mass relative to the case is $ # $ mz std 1 cz std 1 kzstd 5 2my std (3.4.158) where y(t) is the displacement of the case relative to the inertial space. If this displacement is harmonic, y(t)  Y sin t, then by analogy with Eq. (3.4.35) the response is v 2 zstd 5 Y a v b |Gsvd| sin svt 2 fd n 5 Zsvd sin svt 2 fd

(3.4.159)

so that the magnitude factor Z()/Y  (/n) |Gsvd| is as plotted in Fig. 3.4.9 and the phase angle  is as in Fig. 3.4.4. The plot Z()/Y 2

Fig. 3.4.30

versus /n is shown again in Fig. 3.4.30 on a scale more suited to our purposes. Accelerometers are high-natural-frequency instruments. Their usefulness is limited to a frequency range well below resonance. Indeed, for small values of /n, Eq. (3.4.159) yields the approximation Zsvd